Categorical Constructions, Braidings on Monoidal Categories and Bicrossed Products of Hopf Algebras

FACULTEIT WETENSCHAPPEN EN BIO-INGENIEURSWETENSCHAPPEN VAKGROEP WISKUNDE

Categorical constructions, braidings on monoidal categories and bicrossed products of Hopf algebras

Proefschrift voorgelegd aan de Faculteit Wetenschappen, voor het verkrijgen van de graad van Doctor in de Wetenschappen door Ana Loredana Agore

Promotors: Prof. S. Caenepeel Prof. G. Militaru

2012 Contents

Acknowledgements...... iii Introduction...... iv Inleiding...... xiii

1 Categorical constructions for Hopf algebras and related topics1 1.1 Preliminaries...... 1 1.2 Categorical constructions for Hopf algebras...... 14 1.3 Reﬂective and coreﬂective subcategories...... 24 1.4 Monomorphisms of coalgebras and Hopf algebras...... 28 1.5 Braidings on the category of bimodules...... 32 1.6 The center of the category of bimodules...... 43 Bibliographical Notes...... 53

2 Unified products 55 2.1 Notational conventions...... 55 2.2 The extending structures problem...... 56 2.3 Extending structures: the group case...... 57 2.4 Extending structures: the quantum version...... 67 2.5 Bialgebra extending structures and unified products...... 69 2.6 The classification of unified products...... 82 2.7 Unified products and split extensions of Hopf algebras...... 88 2.8 Coquasitriangular structures for extensions of Hopf algebras...... 98

i 2.9 Crossed product of Hopf algebras...... 112 Bibliographical Notes...... 127

3 Classifying bicrossed products of quantum groups. Deformations of a Hopf algebra and descent type theory 129 3.1 Motivating problems...... 129 3.2 The morphisms between two bicrossed products...... 141 3.3 The classiﬁcation of bicrossed products...... 148 3.4 Bicrossed descent theory and deformations of Hopf algebras...... 154 3.5 Examples...... 163 3.6 Application: Bicrossed descent theory for groups...... 184 Bibliographical Notes...... 188

Bibliography 189

ii Acknowledgements

I owe my deepest gratitude to:

Gigel Militaru who suggested most of the problems studied here, for his constant support and guidance along the way. I beneﬁted greatly from his insight on many topics during the scientiﬁc seminar he organized between 2007 and 2010.

Stef Caenepeel for his advice, friendship and hospitality which made my stay in Brussels easier.

The members of the jury: Tomasz Brzezinski,´ Stefaan Caenepeel, Philippe Cara, Eric Jespers, Rudger Kieboom, Gigel Militaru and Joost Vercruysse for the careful reading of this manuscript and their useful suggestions.

My colleagues at the VUB for their friendly attitude and the very nice working environment.

My dear parents and my entire family for their continuous encouragement and support.

Brussel, October 2012

iii Introduction

A bialgebra is an algebra on which there exists a dual structure called a coalgebra such that the two structures are compatible. As in the case of groups where the difference between a group and a monoid relies on the existence of an additional map that allows forming inverses, in the same manner a Hopf algebra is a bialgebra H with an additional linear map S : H → H, called the antipode. However, in this case the antipode provides an inverse for certain linear combinations and not for individual elements. The development of Hopf algebra theory as a distinct branch of mathematics has experienced two historically significant moments: the classical part and the quantum part. The classical part: Hopf algebras were introduced around 1960 arising from algebraic geometry and algebraic topology. In this context Hopf algebras appear either as algebras of regular functions on algebraic groups or as algebras of representative functions on compact Lie groups, or as enveloping algebras of Lie algebras. The first structural results for Hopf algebras motivated by this context are due to A. Borel, J. Dieudonne,´ P. Cartier, G. Hochschild, J. Milnor, J. Moore, B. Kostant, M. Demazure and A. Grothendieck. The first explosion of interest for this field has came in 1969 with Sweedler’s book after which the study of Hopf algebras became a distinct branch of mathematics. Different areas of mathematics like group theory, algebraic groups, algebraic topology, Lie algebras, locally compact groups, representation theory, Galois theory, the theory of graded rings, etc are all intimately related to Hopf algebras. During the classical period the theory of Hopf algebras developed around these ideas, focusing on the attempt to obtain at the level of Hopf algebras general results that extend well known theorems from the above mentioned fields. We mention that even elementary results, such as Lagrange’s theorem in group theory or deep structural connections such as Galois theory became famous conjectures or directions of study for many years in the theory of Hopf algebras. There are two major problems within the classical part of the theory that still generate a lot of interest: on one hand we are talking about the structure and classification of Hopf algebras of a given dimension that satisfy certain properties, whose roots are in the classification of finite groups. On the other hand we are talking about the study of the various types of categories of representations that can be associated with Hopf algebras that (co)act on (co)algebras, whose roots go back to the representation theory of groups, algebras or Lie algebras. The quantum part: the second important development within this field of study started in 1987 with the appearance of the paper Quantum groups by V. Drinfel’d. From this moment on, the field changed dramatically in terms of methods, new examples and interaction with other areas of mathematics such as: noncommutative geometry, physics, knot theory, conformal field theory, category theory, combinatorics, quantum statistical mechanics, etc. For example, noncommutative geometry is based on a simple idea: instead of working with points on a space or a manifold M we can work equivalently with the algebra of functions on M. Adopting a categorical point of view, a quantum space is a representable functor on the category of (not necessarily commutative) algebras. If the representing object of the quantum space is a Hopf algebra then the quantum space is called a quantum group. Quan- tum groups (Hopf algebras) have been therefore accepted as the natural analogue from the point of view of noncommutative geometry of the classical notion of group. A turning point in the theory was the introduction of braided monoidal (tensor) categories by Joyal and Street in 1993. iv The concept proved to be revolutionary by its unifying character and by its implications both in quantum groups as well as in other areas of mathematics: physics, theory of knots, category theory. Braided monoidal categories play a central role in the representation theory of quantum groups, Kac-Moody algebras and quantum field theory. They also provide topological invariants to links, knots and 3-manifolds. Studying objects in braided categories implies working in a very general and unifying framework in mathematics. The present work is devoted to both classical and quantum part of the Hopf algebra theory. In Chapter 1 we study Hopf algebras from the categorical point of view. It is well known that the category k-Alg of k-algebras is complete and cocomplete. This is immediately implied by the existence of products, coproducts, equalizers and coequalizers in the category k-Alg. The categories of coalgebras and bialgebras have arbitrary coproducts and coequalizers (see [58, Propositon 1.4.19], [43, Proposition 2.10], [118, Corollary 2.6.6]), hence these categories are cocomplete. In Section 1.2 we prove that the category of Hopf algebras has coproducts and coequalizers. Furthermore, related to the question of whether these categories are complete (i.e. if they have arbitrary products and equalizers) we could not find similar results in the classical Hopf algebra textbooks ([1], [139]), not even in the more recent ones ([43], [58], [81], [106]). For example, [58, Propositon 1.4.21] proves only the existence of finite products (namely the tensor product of coalgebras) and only in the category of cocommutative coalgebras, as a dual result to the one concerning commutative algebras. We shall fill this gap: using the fact that the forgetful functor from the category of coalgebras to the category of vector spaces has a right adjoint, namely the so called cofree coalgebra, we shall construct explicitly the product of an arbitrary family of coalgebras. As a consequence, the product of an arbitrary family of bialgebras and Hopf algebras is constructed. The equalizers of two morphisms of coalgebras (bialgebras, Hopf algebras) are also described explicitly. Thus we obtain that the categories of coalgebras, bialgebras and Hopf algebras are complete and a description for limits in the above categories is given. Next, we turn our attention to the fundamental book of Sweedler: the following problems concerning Hopf algebras are stated in [139, p. 135], without any proof : given a coalgebra C there exists a free Hopf algebra on C (i.e. the forgetful functor from the category of Hopf algebras to the category of coalgebras has a left adjoint) and a free commutative Hopf algebra on C. The problem has turned out to be quite difficult: several years passed until Takeuchi, in [142, Sec- tions §1 and §11], answered affirmatively to both statements. His proof relies on an ingenious and laborious construction. Moreover, in [139, p. 135] Sweedler also states, again without any proofs, the dual of the above problem: given an algebra A there exists a cofree Hopf algebra on A (that is the forgetful functor from the category of Hopf algebras to the category of algebras has a right adjoint) and a cofree cocommutative Hopf algebra on A. Concerning this problem, recently it was proved in [123, Corollary 4.1.4] that the existence of a cofree Hopf algebra on every algebra implies the existence of a cofree cocommutative Hopf algebra on every algebra. In Section 1.3 we prove Sweedler’s statement concerning the existence of a cofree Hopf algebra on every algebra. For the sake of completeness we also include the construction of this right adjoint which was given in [53]. In Section 1.4 we complete the existing description of monomorphisms in k-Coalg given in [108] with two more characterizations: the first one indicates a cohomological description of

v monomorphisms while the other is an elementary one involving the cotensor product CDC. In any concrete category C the natural problem of whether epimorphisms are surjective maps arise, as well as the dual problem of whether the monomorphisms are injective maps. This type of problems have already been studied before in several well known categories: for example in [133] it is shown that the property of epimorphisms of being surjective holds in the categories of von Neumann algebras, C∗-algebras, groups, finite groups, Lie algebras, compact groups, while it fails to be true in the categories of finite dimensional Lie algebras, semisimple finite dimensional Lie algebras, locally compact groups and unitary rings (see [82], [137]). The more recent paper [53] deals with the same problems in the context of Hopf algebras: several examples of non-injective monomorphisms and non-surjective epimorphisms are given. It turns out that the above problem is also intimately related to Kaplansky’s first conjecture in the way that every non-surjective epimorphism of Hopf algebras provides a counterexample to Kaplansky’s problem. Our interest in this problem comes also from the fact that in the light of [53, Proposition 2.5] which states that a morphism of Hopf algebras is a monomorphism if and only if it is a monomorphism viewed as a morphism of coalgebras it turns out that the same characterization holds for Hopf algebra monomorphisms. Braided monoidal categories play a key role in several areas of mathematics like quantum groups, noncommutative geometry, knot theory, quantum field theory and 3-manifolds. It is well-known that the category AMA of bimodules over an algebra A over a commutative ring k is monoidal. The aim of Section 1.5 is to give an answer to the following natural question: given an algebra A, describe all braidings on AMA. The question is not as obvious as it seems: a first attempt might be to use the switch map to define the braiding, but this is not well-defined, even in the case when A is a commutative algebra. However, there are non-trivial examples of braidings on the category of bimodules. A first general result is Theorem 1.5.1, stating that braidings on the category of A-bimodules are in bijective correspondence with canonical R-matrices, these are invertible elements R in the threefold tensor product A ⊗ A ⊗ A, satisfying a list of axioms. In this situation, we will say that (A, R) is an algebra with a canonical R-matrix. Actually, this result is inspired by a classical result of Hopf algebras: braidings on the category of (left) modules over a bialgebra H are in one-to-one correspondence with quasitriangular structures on H, these are elements R in the two-fold tensor product H ⊗ H satisfying certain properties. We refer to [106, Theorem 10.4.2] for detail. The next step is to reduce the list of axioms to two equations, a centralizing condition and a normalizing condition, and then we can prove in Theorem 1.5.2 that all braidings on a category of bimodules are symmetries. In the situation where A is commutative, we have a complete classification: A admits a canonical R-matrix R if and only if k → A is an epimorphism in the category of rings, and then R is trivial, see Proposition 1.5.3. A The invariants functor G = (−) : AMA → Mk has a left adjoint F = A ⊗ −. We prove that G is a separable functor [109, 131] if and only if G is fully faithful and this implies that A admits a canonical R-matrix. The converse property also holds if A is free as a k-module, and then the braiding on the category of A-bimodules is unique, cf. Theorem 1.5.6. In particular, any Azumaya algebra admits a canonical R-matrix. R can be described explicitly in the cases where A is a matrix algebra or a quaternion algebra, see Examples 1.5.9 and 1.5.10. Not every algebra with a canonical R-matrix is Azumaya; for example Q is not a Z-Azumaya algebra, but 1 ⊗ 1 ⊗ 1 is a canonical R-matrix, since Z → Q is an epimorphism of rings. Thus algebras with vi a canonical R-matrix can be viewed as generalizations of Azumaya algebras. Applying Theorem 1.5.6 to finite dimensional algebras over fields, we obtain a new characterization of central simple algebras, namely central simple algebras are the finite dimensional algebras admitting a canonical R-matrix. As a final application, we construct a simultaneous solution of the quantum Yang-Baxter equation and the braid equation from any canonical R-matrix, see Theorem 1.5.13. A monoidal category can be viewed as a categorical version of a monoid. The appropriate generalization of the center of a monoid is given by the centre construction, which was introduced independently by Drinfeld (unpublished), Joyal and Street [80] and Majid [98]. A key result in the classical theory is the following: the center of the category of representations of a Hopf algebra H is isomorphic to the category of Yetter-Drinfeld modules over H [81]. Moreover, if the Hopf algebra H is finite dimensional, then the category of Yetter-Drinfeld modules is isomorphic to the category of representations over the Drinfeld double D(H). Since the center is a braided monoidal category, it follows that the Drinfeld double is a quasitriangular Hopf algebra. Let A be an algebra over a commutative ring k. In Section 1.6, we study the center of the category AMA of A-bimodules, and relate it to some classical concepts. We introduce A ⊗ Aop-Yetter-Drinfeld modules (Definition 1.6.1), and show that the weak center of AMA is isomorphic to the category of A ⊗ Aop-Yetter-Drinfeld modules (Proposition 1.6.3). We give other descriptions: the weak center is equal to the center (Proposition 1.6.5) and is isomorphic to the category MA⊗A of comodules over the Sweedler canonical coring A ⊗ A (Proposition 1.6.2). Moreover it was proved in [41, Theorem 5] that the category MA⊗A is isomorphic to the category of right A-modules with a flat connection as defined in noncommutative differential geometry. Thus, the category of right A-modules with a flat connection is also equal to the center. We introduce a category of descent data Desc(A/k), generalizing the descent data introduced in [83] from A commutative to A non-commutative, and this category is also isomorphic to the center. The first main result of this section is summarized in Theorem 1.6.9 which provide six isomorphic descriptions for the center of the category of A-bimodules. All six isomorphic categories are symmetric monoidal categories. In particular, the category of comodules over the Sweedler canonical A-coring A⊗A is a symmetric monoidal category and hence one can perform most of the constructions that are performed for differentiable manifolds. For instance, connections in bimodules try to mimic linear connections in geometry and are useful in capturing Riemannian aspects (see [43], [42] for more detalis). The second major application of the above results is the fact that they lead to constructing new and interesting family of solutions for the quantum Yang-Baxter equation (Theorem 1.6.14). Several examples are provided. Chapter 2 deals with the new introduced notion of unified product ([12]). Let C be a category whose objects are sets endowed with various algebraic structures (S) and D be a category such that there exists a forgetful functor F : C → D, i.e. a functor that forgets some of the structures (S). In this context we formulate a general problem which may be of interest for many areas of mathematics: Extending Structures Problem (ES): Let F : C → D be a forgetful functor and consider two objects C ∈ C, D ∈ D such that F (C) is a subobject of D in D. Describe and classify all mathematical structures (S) that can be defined on D such that D becomes an object of C and C is a subobject of D in the category C (the classification is up to an isomorphism that stabilizes

vii C and a certain type of fixed quotient D/C). The ES-problem generalizes and unifies two famous and still open problems in the theory of groups: the extension problem of Holder¨ [73] and the factorization problem of Ore [115]. Let us explain this. In Section 2.3 we formulate the ES-problem at the level of groups, corresponding to the forgetful functor F : Gr → Set: if A is a group and E a set such that A ⊆ E, describe all group structures (E, ·) that can be defined on the set E such that A is a subgroup of (E, ·). In order to do that we have introduced a new product for groups, called the unified product (Theorem 2.3.6), such that both the crossed product (the tool for the extension problem) and the bicrossed product (the tool for the factorization problem) of two groups are special cases of it. The unified product for groups is associated to a group A and a new hidden algebraic structure (H, ∗), connected by two actions and a generalized cocycle satisfying some compatibility conditions. We now take a step forward and formulate the ES-problem at the level of Hopf algebras corresponding to the forgetful functor F : Hopf → CoAlg: (H-C) Extending Structures Problem: Let A be a Hopf algebra and E a coalgebra such that A is a subcoalgebra of E. Describe and classify all Hopf algebra structures that can be defined on E such that A is a Hopf subalgebra of E. If at the level of groups the ES-problem is elementary, for Hopf algebras the problem is more difficult. Indeed, let A be a group and E a set such that A ⊆ E. For a field k we look at the extension k[A] ⊆ k[E], where k[A] is the group algebra that is a Hopf algebra and a subcoalgebra in the group-like coalgebra k[E]. Assume now that (E, ·) is a group structure on the set E such that A is a subgroup of (E, ·). Thus, we obtain an extension of Hopf algebras k[A] ⊆ k[E]. This extension of Hopf algebras has a remarkable property: let H ⊆ E be a system of representatives for the right cosets of the subgroup A in the group (E, ·) such that 1E ∈ H. Since the map u : A × H → E, u(a, h) = a · h is bijective, we obtain that the multiplication map k[A] ⊗ k[H] → k[E], a ⊗ h 7→ a · h is bijective, i.e. the Hopf algebra k[E] factorizes through the Hopf subalgebra k[A] and the subcoalgebra k[H]. This is not valid for arbitrary extensions of Hopf algebras. Therefore, we have to restrict the (H-C) extending structures problem to those Hopf algebras E that factorize through a given Hopf subalgebra A and a given subcoalgebra H: we called this the restricted (H-C) ES-problem and we shall give a complete answer to it in the present chapter. It turns out that H is not only a subcoalgebra of E but will be endowed additionally with a hidden algebraic structure that will play the role of the system of representatives for congruence in the theory of groups. In Section 2.7 we shall prove an equivalent description for the unified product from the view point of split extensions of Hopf algebras. Let A n H be a unified product associated to a bialgebra extending structure Ω(A) = H, /, ., f of a Hopf algebra A (see Section 3.1). Then we have an extension of bialgebras iA : A → A n H. This extension is split in the sense of [136]: there exists πA : A n H → A a left A-module coalgebra morphism such that πA ◦ iA = IdA. Thus the unified product A n H appears as a special case of the Schauenburg’s product A ∝ H but is a much more malleable version of it. Furthermore, there is more to be said and this makes a substantial difference: πA is also a normal morphism of coalgebras in the sense of Andruskiewitsch and Devoto [21]. This context fully characterizes unified products: we prove that a Hopf algebra E is isomorphic to a unified product A n H if and only if there viii exists a morphism of Hopf algebras i : A → E which has a retraction π : E → A that is a normal left A-module coalgebra morphism (Theorem 2.7.3). The next aim of this section is to make the connection between the unified product and Radford’s biproduct: for a Hopf algebra A, Proposition 2.7.6 gives necessary and sufficient conditions for iA : A → A n H to be a split monomorphism of bialgebras. In this case the unified product A n H is isomorphic as a A bialgebra to a biproduct L ∗ A, and the structure of L as a bialgebra in the category AYD of Yetter-Drinfel’d modules is explicitly described. Finally, Theorem 2.7.8 gives a general method for constructing unified products, as well as biproducts arising from a right A-module coalgebra (H, C) and a unitary coalgebra map γ : H → A. In Section 2.8 we completely describe the coquasitriangular structures on unified products (see Theorem 2.8.6). Let λ : H ⊗ A → k be a skew pairing between two Hopf algebras and consider Dλ(A, H) := A ./λ H to be the generalized quantum double as constructed in ([100, Example 7.2.6]). In particular, the coquasitriangular structures on a bicrossed product are given in Theorem 2.8.8. As the main application of Theorem 2.8.8 the set of all coquasitriangular structures on the generalized quantum double Dλ(A, H) are completely described. In particular, it is proved that a generalized quantum double is a coquasitriangular Hopf algebra if and only if both Hopf algebras A and H are coquasitriangular. Several explicit examples are also provided. Section 2.9 deals with the crossed product of Hopf algebras. The crossed product is a fundamental construction in mathematics. It was first introduced in group theory related to the famous extension problem of Holder:¨ any extension (E, i, π) of a group H by a group G is equiva- α lent to a crossed product extension (H#f G, iH , πG). The construction of crossed products of groups has served as a model for later generalizations at the level of groups acting on rings [122], Hopf algebras acting on k-algebras [35], von Neumann algebras [107], etc. The crossed product . A#f H of a Hopf algebra H acting on a k-algebra A was introduced independently in [35] and [62] as a generalization of the crossed product of groups acting on k-algebras. It has only an algebra structure and was studied mainly as an algebra extension of A, being an essential tool in Hopf-Galois extensions theory as it is well known that Hopf-Galois extensions with normal basis are equivalent to crossed products with invertible cocycle. Many algebraic properties of the crossed product of a Hopf algebra H acting on a k-algebra A such as semisimplicity, semiprimeness, etc. were studied in this setting ([52], [127]). If, in addition, A and H are Hopf algebras and the cocycle f : H ⊗ H → A and the action . : H ⊗ A → A are coalgebra maps satisfying two compatibility conditions then we proved in Example 2.5.6, 2) that the crossed product . A#f H has a natural Hopf algebra structure which we call the crossed product of Hopf algebras. An important feature of the crossed product of Hopf algebras is the following: a Hopf algebra E is isomorphic as a Hopf algebra to a crossed product of Hopf algebras if and only if E factorizes through a normal Hopf subalgebra and a subcoalgebra containing the unit of E (Theorem 2.9.3). The aim of Chapter 3 is to prove that there exists a very rich theory behind the so-called bicrossed product (or double cross product in Majid’s terminology) of two objects which deserves to be developed further mainly because of the major impact they have in at least three different problems: the classification of objects of a given dimension, the development of a general descent type theory for a given extension A ⊆ E (including a deformation type theory as a subsequent problem) which we called bicrossed descent theory as the converse of the factorization problem and also the development of some new types of cohomologies that will be the key players

ix for both problems. All results presented below provide a detailed answer at the level of Hopf algebras for the three problems mentioned above and offer an argument for the major role that bicrossed products can play. In particular, for finite quantum groups we describe a new way of approaching the classification problem which we hope to be effective in the future. In order to maintain a general frame for our discussion, considering that bicrossed products were introduced and studied in various areas of mathematics, we will consider C a category whose objects are sets endowed with various algebraic, topological, differential structures. To illustrate, we can think of C as the category of groups, groupoids or quantum groupoids, algebras, Hopf algebras, local compact groups or local compact quantum groups, Lie groups, Lie algebras and so on. Let A and H be two given objects of C. We say that an object E ∈ C factorizes through A and H if E can be written as a ’product’ of A and H, where A and H are viewed as subobjects of E having minimal intersection. Here, the ’product’ depends on the nature of the category. For instance, if C = Gr, the category of groups, then a group E factorizes through two subgroups A and H if E = AH and A ∩ H = {1}. This is called in group theory an exact factorization of E and can be restated equivalently as the fact that the multiplication map A×H → E, (a, h) 7→ ah is bijective. The last assertion is also taken as a definition of factorization for other categories like: algebras [49], Hopf algebras [100], Lie groups or Lie algebras [96], [84], [103], locally compact quantum groups [145] and so on. The factorization problem is then the following very natural and elementary question: The factorization problem. Let A and H be two given objects of C. Describe and classify up to an isomorphism all objects of C that factorize through A and H. There is also an interesting converse of the above problem, called here the bicrossed descent theory, which we introduce below having as main source of inspiration the classical descent theory for modules [83]. Now let A ⊂ E be two given objects of C such that A is an subobject of E.A factorization A-form of E is a suboject H of E such that E factorizes through A and H. We denote by F(A, E) the (possibly empty) full subcategory of C of all factorization A-forms of E. The bicrossed descent theory consists of the following two questions: Existence of forms. Let A ⊂ E be an extension in C. Does there exist a factorization A-form of E? Description and classification of forms. If a factorization A-form of E exists, describe and classify up to isomorphism all factorization A-forms of E. Going back to the factorization problem for groups, an important step in dealing with this problem was the construction of the bicrossed product A ./ H associated to a matched pair (A, H, C, B) of groups given by Takeuchi [143]. A group E factorizes through two subgroups ∼ A and H if and only if there exists a matched pair of groups (A, H, C, B) such that E = A ./ H. Thus the factorization problem can be restated in a pure computational manner: Let A and H be two given groups. Describe the set of all matched pairs (A, H, C, B) and classify up to an isomorphism all bicrossed products A ./ H. In conclusion, if we are only looking for the description part of the factorization problem, formulated in an arbitrary category C, the following general principle (for the categories mentioned above this principle becomes a theorem) should work: an object E ∈ C factorizes through A

x and H if and only if E =∼ A ./ H, where A ./ H is a ’bicrossed product’ in the category C associated to a ’matched pair’ between the objects A and H. The classification part of the factorization problem is now clear: it consists of classifying the bicrossed products A ./ H associated to all matched pairs between A and H. This is the strategy that we follow for the category of Hopf algebras. For other categories, the steps taken into this direction are still shy, including the group case as well as the algebras case presented above. The present chapter offers an answer to the first and the third problem above if C = Hopf, the category of Hopf algebras. The second problem, namely the existence of forms, needs to be treated ”case by case” for every given Hopf algebra extension A ⊆ E, a computational part of it can not be avoided. The chapter is organized as follows. In Section 3.1 we shall recall the basic concepts that will be used throughout the chapter. Section 3.2 is devoted to proving some purely technical results which will be intensively used throughout the chapter. Theorem 3.2.2 describes completely the set of all morphisms of Hopf algebras ψ : A ./ H → A0 ./0 H0 between two arbitrary bicrossed products. In particular, the set of all Hopf algebra maps between two semi-direct (or smash) products of Hopf algebras is described in Corollary 3.2.3. Section 3.3 deals with the classification part of the factorization problem for which the group 1 Hl,c(H,A) of all coalgebra lazy 1-cocycles of H with coefficients in A introduced in Defini- tion 2.6.3 plays the crucial role. Let A and H be two given Hopf algebras. Theorem 3.3.7 is the classification theorem for bicrossed products: all Hopf algebras E that factorize through A and H are classified up to an isomorphism that stabilizes A by a cohomological type object 2 1 H (A, H) in the construction of which the key role is played by pairs (r, v) ∈ Hl,c(H,A) × 1 Aut CoAlg(H), consisting of a coalgebra lazy 1-cocycle r : H → A and an unitary automorphism of coalgebras v : H → H related by a certain compatibility condition. The classification of bicrossed products up to an isomorphism that stabilizes one of the terms has at least two strong motivations: the first one is the cohomological point of view which descends to the classification theory of Holder’s¨ group extensions [133, Theorem 7.34] and the second one is the problem of describing and classifying the A-forms of a Hopf algebra from descent theory. Section 3.4 offers the full answer to the third problem above on the description and classification of forms as part of what we have called the bicrossed descent theory. The answer will be given in four steps, each of them of interest in its own right, as follows: deformation of a Hopf algebra, deformations of forms, the description of forms and finally the classification of forms. In Theorem 3.4.7 a general deformation of a given Hopf algebra H is introduced. This deformation Hr of H is associated to an arbitrary matched pair of Hopf algebras (A, H, B, C) and to an (B, C)-cocycle r : H → A in the sense of Definition 3.4.4. As a coalgebra Hr = H, with the new multiplication • defined by h • g := h / r(g(1)) g(2) for all h, g ∈ Hr = H. Then Hr is a new Hopf algebra called the r-deformation of H. Now let A ⊆ E be an extension of Hopf algebras and F(A, E) the small category, possibly empty, of all factorization A-forms of E: hence, F(A, E) is the category of all Hopf subalgebras H ⊆ E such that E factorizes through A and H. Let F sk(A, E) be the skeleton of F(A, E), that is a set of types of isomorphisms of all factorization A-forms of E. The factorization index of the extension E/A, introduced in Definition 3.4.2 and denoted by [E : A]f , is the cardinal of F sk(A, E), i.e. [E : A]f = | F sk(A, E) |. The extension A ⊆ E is called rigid if [E : A]f = 1.

xi Rigid extensions are interesting for the following reason. Assume that E/A is rigid: if E =∼ A ./ H =∼ A ./0 H0, then H =∼ H0. This is a Krull-Schmidt-Azumaya type theorem for bicrossed product of two Hopf algebras: the rigid extensions of Hopf algebras are exactly those for which the decomposition as a bicrossed product is unique. Examples of rigid extensions as well as of extensions E/A such that [E : A]f ≥ 2, which are quite rare, are provided. Theorem 3.4.9 proves that if r : H → A is an (B, C)-cocycle, then the r-deformation Hr is a factorization r A-form of the bicrossed product A ./ H, that is there exists a new matched pair (A, Hr, B , C) r ∼ such that A ./ Hr = A ./ H. We called this result deformation of forms: another name used for a similar result at the level of algebras is invariance under twisting theorem [76, Theorem 4.4]. The description of forms is given in Theorem 3.4.10 which is the converse of Theorem 3.4.9: if H is a given factorization A-form of E then any other form H is isomorphic as a Hopf algebra with an Hr, for some (B, C)-cocycle r : H → A. This result is interesting in its own right as it proves that in order to find all the objects of the category F(A, E) of all factorization A-forms of E it is enough to know only one object H: all other objects are deformations of H. Finally, as a conclusion of these theorems, the classification of forms is proved as the main result of the section, namely Theorem 3.4.6: if H is a given factorization A-form of E then there exists a bijection between the set of types of isomorphisms of all factorization A-forms of E and a new 2 cohomological object HA (H,A | (B, C)). In particular, we obtain a formula for computing the f 2 factorization index of a given extension A ⊆ E: [E : A] = |HA (H,A | (B, C))|. Mutatis- mutandis, Theorem 3.4.6 can be viewed as a bicrossed version for Hopf algebras of the classical result in descent theory: if k ⊆ l is a faithfully flat extension of commutative rings then the first Amitsur cohomology group is isomorphic to the relative Picard group Pic(l/k) ([83]). In Section 3.5 we provide some explicit examples: for two given Hopf algebras A and H we will describe by generators and relations and classify up to an isomorphism all Hopf algebras E that factorize through A and H. Furthermore, for any such Hopf algebra E the factorization index [E : A]f is computed. There are three steps that we go through: first of all we compute the set of all matched pairs between A and H. This is the computational part of our schedule and can not be avoided. Then we describe by generators and relations the bicrossed products A ./ H associated to all these matched pairs. Finally, using Theorem 3.2.2, we shall classify up to an isomorphism these bicrossed products A ./ H. As an application, the group Aut Hopf (A ./ H) of all Hopf algebra automorphisms of a given bicrossed product is computed. All the results proved in Section 3.4 can be translated to groups by replacing the category C = Hopf with the category Gr of groups, without needing a proof. We write down briefly these new results in Section 3.6.

xii Inleiding

Een bialgebra is een lineaire ruimte voorzien van een algebra en een coalgebra structuur die met elkaar compatibel zijn. Een Hopf algebra is een bialgebra H voorzien van een lineaire afbeelding S : H → H, genaamd de antipode met eigenschappen die kunnen vergeleken worden met de eigenschappen van inverse elementen, die van een monoide een groep maken. Historisch kunnen we twee belangrijke bloeiperioden onderscheiden in de ontwikkeling van de algebraische theorie van Hopf algebras, die we de klassieke periode en de quantum periode kunnen noemen. Klassieke periode: De oorsprong van Hopf algebras vinden we terug in algebraische topologie en algebraische meetkunde, vanaf de jaren 40. Een doorgedreven algebraische studie situeert zich in de periode 1960-1980; Hopf algebras manifesteren zich als algebras of reguliere functies op algebraische groepen, of als algebras van representerende functies on compacte Lie groepen, of als omhullend algebras van Lie algebras. De eerste belangrijke resultaten in dit verband zijn toe te schrijven aan Borel, Dieudonne,´ Cartier, Hochschild, Milnor, Moore, Kostant, De- mazure en Grothendieck. Het hoogtepunt van de eerste bloeiperiode is wellicht de publicatie van Sweedler’s monografie “Hopf algebras” in 1969; vanaf dan kunnen we Hopf algebra theorie beschouwen als een aparte wiskundige discipline. Een reeks wiskundige gebieden, zoals groepentheorie, algebraische groepen, algebraische topologie, Lie algebras, locaal compacte groepen, representatie theorie, Galois theorie, gegradeerde ringen zijn nauw verweven met Hopf algebras. Het is rond deze ideeen¨ dat de theorie van Hopf algebras zich ontwikkeld heeft tijdens de eerste bloeiperiode; hierbij lag de focus in veel gevallen op veralgemeningen en unificaties van resultaten uit bovenvermelde disciplines naar het meer algemene kader van Hopf algebras. Laat ons hier vermelden dat een aantal elementaire resultaten, zoals de stelling van Lagrange in groepentheorie, of de klassieke Galois theorie, hebben aanleiding gegeven tot problemen over Hopf algebras die pas na vele jaren opgelost werden. Twee problemen worden ook vandaag nog intensief bestudeerd. Ten eerste is er het classificatieprobleem, dat teruggaat tot de classificatie van eindige groepen: klasseer Hopf algebras van een gegeven dimensie, al dan niet met bepaalde vooraf gegeven eigenschappen. Daarnaast is er de studie van acties en coacties van Hopf algebras op algebras en coalgebras, een probleem dat zijn wortels vindt in de klassieke rep- resentatietheorie van groepen, algebras en Lie algebras. Quantum periode. De tweede bloeiperiode nam een aanvang in 1987, met de publicatie van het artikel “Quantum groups” door Victor Drinfeld, dat een revolutie teweegbracht: nieuwe technieken, nieuwe voorbeelden, en een vloed aan nieuwe interacties met andere wiskundige disciplines, zoals niet-commutatieve meetkunde, wiskundige natuurkunde, knopentheorie, conformele veldentheorie, categorie theorie, combi- natoriek en statistische quantum mechanica. Niet-commutatieve meetkunde, bijvoorbeeld, is essentieel gebaseerd op een eenvoudig idee. Werken met punten in een ruimte of een manifold M is equivalent met het werken met de algebra van functies op M. Vanuit een categorisch standpunt is een quantum ruimte een representeerbare functor op de categorie van (niet noodzakelijk commutatieve) algebras. Als het representerende object een Hopf algebra is, dan noemen we deze quantum ruimte een quantum groep. Quantum groepen (of Hopf algebras) zijn nu aan- vaard als het natuurlijke analogon van klassieke groepen, tenminste vanuit een niet-commutatief meetkundig standpunt.

xiii Een nieuw keerpunt was de invoering van braided (gebreide) monoidale categorieen¨ door Joyal en Street in 1993. Dit werk was een mijlpaal, door het uniﬁcerende karakter ervan, en door de gevolgen ervan, niet alleen op de theorie van de quantum groepen, maar ook op categorie theorie, knopen theorie, en wiskundige fysica. Ons werk behelst aspecten van zowel het klassieke als het quantum deel van de Hopf algebra theorie.

In Hoofdstuk 1 bestuderen we Hopf algebras vanuit een categorisch standpunt. Het is welbekend dat de categorie van de algebras volledig en co-volledig is, een gevolg van het bestaan van producten, coproducten, equalizers en coequalizers van algebras. De categorieen¨ van de coalgebras en van de bialgebras hebben willekeurige coproducten en coequalizers (zie [58, Proposition 1.4.19], [43, Proposition 2.10], [118, Corollary 2.6.6]), en dus zijn ze co-volledig. In § 1.2 zullen we aantonen dat ook de categorie van de Hopf algebras coproducten en coequalizers heeft. Een natuurlijke vraag is of deze categorieen¨ ook compleet zijn; in de klassieke literatuur over Hopf algebras hebben we geen antwoord op deze vraag kunnen vinden. Een zeer gedeeltelijk resultaat kan gevonden worden in [58, Proposition 1.4.21], waar het bestaan van eindige producten (dit zijn hier tensor producten) van cocommutatieve coalgebras bestaan. Dit is een lacune die we zullen opvullen. Gebruik makend van het resultaat dat de vergeetfunctor van coalgebras naar vectorruimten een rechtstoegevoegde heeft, de zogenaamde co-vrije coalgebra, kunnen we expliciet het product van een willekeurige familie coalgebras construeren. Als toepassing kunnen we ook het product van een willekeurige familie bialgebras of Hopf algebras beschrijven. Equal- izers van families coalgebras, bialgebras en Hopf algebras kunnen we ook expliciet beschrijven, en dus zijn de bijhorende categorieen¨ compleet. We geven ook een expliciete beschrijving van limieten in deze categorieen.¨ In het basiswerk van Sweedler vinden we volgende problemen, zonder enig bewijs, zie [139, p. 135]. Bestaat er een vrije Hopf algebra over een coalgebra C? Anders geformuleerd, heeft de vergeetfunctor van Hopf algebras naar coalgebras een linkstoegevoegde? Eenzelfde vraag kunnen we stellen voor de cateogrie van commutatieve Hopf algebras. Het heeft vele jaren geduurd tot Takeuchi een positief antwoord heeft gegeven op beide vragen, in [142, Sections §1 and §11] Zijn bewijs is gebaseerd op een ingenieuze constructie. Ook het duale probleem wordt reeds geformuleerd in het boek van Sweedler, zie [139, p. 135]: bestaat er een covrije Hopf algebra over een algebra A? Anders geformuleerd: heeft de vergeetfunctor van Hopf algebras naar algebras een rechtstoegevoegde? Kort geleden werd bewezen dat het bestaan van covrije Hopf algebras over algebras het bestaan impliceert van covrije cocommutatieve Hopf algebras over algebras, zie [123, Corollary 4.1.4]. We zullen het bestaan van covrije Hopf algebras over algebras aantonen in § 1.3. Voor de volledigheid geven we ook de constructie van de rechtstoegevoegde, zoals beschreven in [53]. Monomorfismen in k-Coalg werden gekarakteriseerd in [108]; in § 1.4 voegen we daar nieuwe karakterisaties aan toe; een eerste karakterizatie is cohomologisch, en de tweede houdt verbandt met het cotensor product CDC. In een concrete categorie C kan de natuurlijke vraag gesteld worden of epimorfismen samenvallen met surjecties, en, duaal, monomorfismen met injecties. Dit type problemen is veelvuldig bestudeerd geweest. In [133] wordt aangetoond dat epimorfismen surjecties zijn in de volgende categorieen:¨ von Neumann algebras, C∗-algebras, groepen, eindige groepen, Lie algebras, compacte groepen. De eigenschap geldt echter niet in de cate- xiv gorieen¨ der eindig dimensionale Lie algebras, semienkelvoudige eindig dimensionaleLie algebras, locaal compacte groepen en ringen met een eenheid, zie [82], [137]. In [53] wordt hetzelfde probleem bestudeerd in de context van Hopf algebras; hier worden diverse voorbeelden van niet- injectieve monomorfismen en niet-surjectieve epimorfismen gegeven. Het is ook gebleken dat dit probleem nauw verband houdt met eerste vermoeden van Kaplansky, in die zin dat elk niet- surjectief epimorfisme van Hopf algebras een tegenvoorbeeld voor Kaplansky’s vermoeden met zich meebrengt. Onze interesse in dit probleem vloeit ook voor uit [53, Proposition 2.5]: een Hopf algebra morfisme is een monomorfisme als en alleen als het een monomorfisme is in de categorie der coalgebras. Gebreide monoidale categorieen¨ spelen een sleutelrol in diverse gebieden binnen de wiskunde: quantum groepen, niet-commutatieve meetkunde, knopentheorie, quantum veldentheorie en 3- manifolds. Het is welbekend dat de categorie der bimodulen AMA over een algebra A monoidaal is. We doel van § 1.5 is een antwoord te geven op de volgende vraag: beschrijf alle gebreide structuren op AMA. Deze vraag is niet zo eenvoudig als ze lijkt, zelfs in het geval waarin A commutatief is. Toch bestaan er niet-triviale voorbeelden. Een eerste algemeen resultaat is hoofdstelling 1.5.1: gebreide structuren op de categorie der bimodulen corresponderen bijectief met kanonieke R-matrices, dit zijn inverteerbare elementen R in het drievoudige tensor product A ⊗ A ⊗ A, die aan zekere axioma’s voldoen. We zullen dan zeggen dat (A, R) een algebra met een kanonieke R-matrix is. Dit resultaat is trouwens geinspireerd door een klassiek resultaat over Hopf algebras, namelijk het feit dat gebreide structuren op de categorie der linkse modulen over een bialgebra H bijectief corresponderen met quasitriangulaire structuren op H, dit zijn elementen in het tweevoudig tensor product H ⊗ H, die aan zekere voorwaarden voldoen, zie bijvoorbeeld [106, Theorem 10.4.2]. De lijst van axioma’s waaraan R moet voldoen kan gereduceerd worden to twee vergelijkingen, namelijk een centralizerende en een normaliz- erende voorwaarde. In hoofdstelling 1.5.2 bewijzen we dat alle gebreide structuren symmetrieen¨ zijn; in het geval waarin A commutatief is hebben we een volledige karakterizatie: A heeft een kanonieke R-matrix als en alleen als k → A een epimorfisme is in de categorie der ringen, en in dit geval is R triviaal, zie stelling 1.5.3. A De invarianten functor G = (−) : AMA → Mk heeft een linkstoegevoegde F = A ⊗ −. We toenen aan dat G separabel is in de zin van [109, 131] als en alleen als G voltrouw is, en dit impliceert dat A kan uitgerust worden met een voltrouwe R-matrix. De omgekeerde eigenschap geldt wanneer A vrij is als een K-moduul, en dan is de gebreide structuur uniek, zie hoofdstelling 1.5.6. In het bijzonder heeft elke Azumaya algebra een kanonieke R-matrix. R kan expliciet beschreven worden in de situaties waarin A een matrix algebra of een quaternionen algebra is, zie voorbeelden 1.5.9 and 1.5.10. Niet elke algebra met een kanonieke R-matrix is Azumaya; Q is niet Azumaya als een Z-algebra, maar heeft wel een kanonieke R-matrix. Als we hoofdstelling 1.5.6 toepassen op eindig dimensionale algebras, dan verkrijgen we een alternatieve karakterizatie van centraal enkelvoudige algebras, het zijn namelijk precies de eindig dimensionale algebras die een kanonieke R-matrix hebben. Een verdere toepassing is de constructie van een gelijktijdige oplossing van de quantum Yang-Baxter vergelijking en de brei vergelijking, vertrekkende van een kanonieke R-matrix, zie hoofdstelling 1.5.13. Een monoidale categorie is in wezen de categorische versie van een monoide. De centrum constructie is de categorische versie van het centrum van een monoide; deze werd onafhankelijk

xv van elkaar ingevoerd door verschillende auteurs: Drinfeld (niet gepubliceerd), Joyal en Street [80] en Majid [98]. Een cruciaal resultaat is dat het centrum van de categorie der representaties van een Hopf algebra H isomorf is met de categorie der Yetter-Drinfeld modulen over H, zie bijvoorbeeld [81]. Als H ook eindigdimensionaal is, dan is het centrum ook isomorf met de categorie der representaties van de Drinfeld dubbel D(H). Omdat het centrum bij constructie een gebreide monoidale categorie is, volgt hieruit dat de Drinfeld dubbel een quasitriangulaire Hopf algebra is. In § 1.6 bestuderen we het centrum van de categorie AMA der bimodulen over een k-algebra A. We voeren A ⊗ Aop-Yetter-Drinfeld modulen in, zie Definition 1.6.1, en laten op zien dat het zwakke centrum van AMA isomorf is met de categorie der A ⊗ A -Yetter-Drinfeld modulen, zie stelling 1.6.3. We geven ook andere beschrijvingen: het zwakke centrum is gelijk aan het centrum, en isomorf met de categorie MA⊗A bestaande uit comodulen over Sweedler’s kanonieke coring. In [41, Theorem 5] werd bewezen dat deze ook isomorf is met de categorie der A-modulen met een platte connectie, zoals gedefinieerd in niet-commutatieve algebraische meetkunde. We voeren ook een categorie Desc(A/k) van “descent data” in, en deze is een niet- commutatieve veralgemening van de descent data ingevoerd in [83], en deze is ook isomorf met het centrum. De resultaten worden samengevat in hoofdstelling 1.6.9, waarin zes categorieen¨ worden beschreven die isomorf zijn met het centrum. Deze zijn allen symmetrisch, en in het bijzonder is de categorie der comodulen over Sweedler’s kanonieke coring symmetrisch, een eigenschap die voorheen onbekend was, en ondertussen reeds toepassingen heeft, zie [42]. Onze constructies hebben ook geleid tot nieuwe interessante oplossingen van de quantum Yang-Baxter vergelijking, zie hoofdstelling 1.6.14. We presenteren diverse voorbeelden. In Hoofdstuk 2 bestuderen we eengemaakte producten, zie [12]. Zij C een categorie waarvan de objecten verzamelingen zijn, uitgerust met een algebraische structuur (S), en D een categorie zodanig dat er een vergeetfunctor F : C → D bestaat. We kunnen dan het volgende algemeen probleem formuleren. Uitbreiden van de structuur (ES): Beschouw C ∈ C, D ∈ D, zodat F (C) een deelobject is van D. Beschrijf en klasseer alle mogelijke structuren (S) die kunnen gedefinieerd worden op D zodat D ∈ C en C is een deelobject van D in C. Het ES-probleem veralgemeent twee bekende problemen uit de groepentheorie: het extensie probleem van Holder¨ [73] en het factorizatie probleem van Ore [115]. In § 2.3 zullen we het ES-probleem formuleren voor groepen, meer bepaald voor de vergeetfunctor F : Gr → Set: als A een groep is, en E een verzameling die A bevat, beschrijf dan alle groepstructuren (E, ·) zodat A een deelgroep is van E. Om dit te kunnen doen moeten we een nieuw product van groepen invoeren, genaamd het eengemaakt product, zie hoofdstelling 2.3.6, waarvan zowel het gekruist product (hulpmiddel voor het extensie probleem) en het dubbelgekruist product (hulpmiddel voor het factorizatieprobleem) speciale gevallen zijn. Het eengemaakt product kunnen we associeren aan een groep A en een nieuwe algebraische structuur (H, ∗), aan elkaar verbon- den via twee acties, en een veralgemeend cocycle dat aan zekere compatibiliteitscondities moet voldoen. De volgende stap bestaat er nu in om het ES-probleem te formuleren op het niveau van Hopf algebras, voor de vergeetfunctor F : Hopf → CoAlg. (H-C) Uitbreiden van de structuur: Beschouw een Hopf algebra A en een coalgebra E zodanig dat A een deelcoalgebra is van E. Beschrijf en klasseer alle Hopf algebra structuren op

xvi E zodat A een Hopf deelalgebra is van E. Dit probleem is ingewikkelder dan het corresponderend probleem voor groepen. Neem een groep A en een verzameling E zodat A ⊂ E. We kunnen dan kijken naar de uitbreiding k[A] ⊆ k[E]. Onderstel nu dat (E, ·) een groepsstructuur is op E zodat A een deelgroep is van E. We hebben dan een Hopf algebra uitbreiding k[A] ⊆ k[E], en deze heeft een opmerkenswaardige eigenschap die in het algemeen niet geldt. Neem een representerend systeem H ⊆ E voor de rechtse cosets van de deelgroep A van (E, ·), zodat 1E ∈ H. De afbeelding u : A × H → E, u(a, h) = a · h is bijectief, en dus is ook de vermenigvuldiging

k[A] ⊗ k[H] → k[E], a ⊗ h 7→ a · h bijectief, wat betekent dat de Hopf algebra k[E] factorizeert door de Hopf deelalgebra k[A] en de deelcoalgebra k[H]. Zoals gezegd geldt dit niet voor willekeurige extensies van Hopf algebras, en hierdoor moeten we het extensie probleem (H-C) beperken tot die Hopf algebras E die factorizeren door een gegeven Hopf deelalgebra A en een gegeven coalgebra H. Dit probleem noemen we het beperkte (H-C) ES-probleem, en in Hoofdstuk 2 geven we een volledig antwoord hierop. Het blijkt dat H niet alleen een deelcoalgebra is van E, maar ook een verborgen algebra structuur heeft, die de rol speelt van het systeeem van represantanten in de theorie voor extensies van groepen. In § 2.7 geven we een equivalente beschrijving van het eengemaakt product, vanuit het standpunt van gespleten uitbreidingen van Hopf algebras. Neem een eengemaakt product A n H, geassocieerd aan de bialgebra structuur Ω(A) = H, /, ., f , zie § 3.1. We hebben dan een bialgebra uitbreiding iA : A → A n H, en deze is gespleten in de zin van [136]: er bestaat een links A-moduul coalgebra morfisme πA : A n H → A zodat πA ◦ iA = IdA. Het eengemaakt product is dus een speciaal geval van het Schauenburg product A ∝ H, maar er valt wel gemakkelijker mee te werken. Bovendien is πA een normaal morfisme van coalgebras, in de zin van Andruskiewitsch and Devoto [21]. Dit is een volledige karakterizatie van eengemaakte producten: een Hopf algebra E is isomorf met een eengemaakt product A n H als en slechts als er een morfisme van Hopf algebras i : A → E, met een retractie π : E → A die een normaal links A-moduul coalgebra morfisme is, zie hoofdstelling 2.7.3. Het volgende doel is om een verband te leggen tussen het eengemaakte product en het Radford biproduct: in stelling 2.7.6 geven we nodige en voldoende voorwaarden opdat iA : A → A n H een gespleten monomorfisme van bialgebras is. In dit geval is AnH isomorf als bialgebra met een biproduct L∗A, en de structuur A van L als bialgebra in de categorie AYD der Yetter-Drinfeld modulen can expliciet beschreven worden. In hoofdstelling 2.7.8 geven we een algemene methode om eengemaakte producten te construeren, en biproducten die voortkomen uit een rechtese A-moduul coalgebra (H, C) en een unitaire coalgebra afbeelding γ : H → A. In § 2.8 geven we een volledige beschrijving van coquasitriangulaire structuren op eengemaakte producten, zie hoofdstelling 2.8.6. Zij λ : H ⊗ A → k een scheve paring tussen twee Hopf algebras, and beschouw de veralgemeende quantum dubbel Dλ(A, H) := A ./λ H zoals die geconstrueerd werd in [100, Example 7.2.6]. In hoofdstelling 2.8.8 geven we de coquasitriangulaire structuren op een dubbel gekruist product. Als toepassing kunnen we de coquasitriangulaire structuren op Dλ(A, H) volledig beschrijven. In het bijzonder kunnen we aantonen dat een veralgemeende quantum dubbel coquasitriangulair is als en slechts als beide onderliggende Hopf algebras A and H coquasitriangulair zijn. We geven ook een aantal voorbeelden.

xvii In § 2.9 bestuderen we gekruiste producten van Hopf algebras. Gekruiste producten werden eerst ingevoerd in groep theorie, naar aanleiding van het extensie probleem van Holder:¨ een uitbreiding (E, i, π) van een groep H door een groep G is equivalent met een gekruiste product uit- α breiding (H#f G, iH , πG). Deze constructie heeft later als model gediend bij de invoering van gekruiste producten in het kader van groepacties op ringen [122], Hopf algebra acties op algebras . [35], von Neumann algebras [107], enz. Het gekruiste product A#f H van een Hopf algebra H agerend op een algebra A werd onafhankelijk ingevoerd in [35] en [62] als een veralgemening van de situatie waarbij een groep ageert op een algebra. Dit gekruist product heeft enkel een algebra structuur, en het werd bestudeerd als algebra uitbreiding van A. Het gekruist product is essentieel in Hopf-Galois theorie: Hopf-Galois uitbreidingen met normale basis zijn precies gekruiste producten met een inverteerbare cocycle. Algebraische eigenschappen van gekruiste producten, zoals halfenkelvoudigheid, halfpriemheid, en anderen werden bestudeerd in onder- meer [52], [127]. In voorbeeld 2.5.6 bewijzen we dat het gekruiste product een natuurlijke Hopf algebra structuur draagt, als aan bijkomend voorwaarden voldaan is. We noemen dit het gekruist product van Hopf algebras. Een opmerkelijk resultaat is dat een Hopf algebra E isomorf is met een gekruist product van Hopf algebras als en slechts als E factorizeert door een normale Hopf deelalgebra en een deelcoalgebra die de eenheid van E bevat, zie hoofdstelling 2.9.3. Het doel van Hoofdstuk 3 is aan te tonen dat er een rijke theorie verscholen zit achter dubbel gekruiste producten; deze heeft een impact op tenminste drie algemene problemen: classificatie van objecten van een gegeven dimensie; ontwikkeling van een algemene “descent” theorie voor een gegeven uitbreiding A ⊆ E, met inbegrip van deformatie achtige theorieen),¨ en de ontwikkeling van nieuwe types van cohomologieen.¨ Antwoorden op deze drie problemen worden gegeven in het kader van Hopf algebras, en het zal blijken dat dubbel gekruiste producten hierin een cruciale rol spelen. In het bijzonder stellen we een alternatieve benadering voor het classificatie probleem voor eindige quantum groepen voor, en we koesteren de hoop dat die in de toekomst efficient¨ zal blijken te zijn. Om een algemeen raamwerk voor onze discussie te kunnen aanhouden, in acht genomen de rol die dubbel gekruiste producten spelen in diverse takken van de wiskunde, werken we in een categorie C waarvan de objecten verzamelingen zijn voorzien van diverse algebraische, topologische of differentiele¨ structuren, zoals groepen, quantum groepoiden, locaal compacte groepen, Lie groepen of Lie algebras. Neem twee objecten A en H in C. We zeggen dat E ∈ C factorizeert door A en H als E kan geschreven worden als een “product” van A en H, waar A en H bekeken worden als deelobjecten van H met een minimale doorsnede. Het product hangt hier af van de aard van de categorie. In het geval waarin C = Gr, de categorie van de groepen, hebben we bijvoorbeeld dat E factorizeert door twee deelgroepen A en H als E = AH en A ∩ H = {1}. In groepentheorie wordt dit een exacte factorizatie genoemd, en een equivalente formulering is de voorwaarde dat de vermenigvuldiging A × H → E, (a, h) 7→ ah bijectief is. Deze laatste voorwaarde wordt ook genomen als definitie voor factorizatie in andere categorieen¨ zoals algebras [49], Hopf algebras [100], Lie groups en Lie algebras [96], [84], [103], locaal compacte quantum groupen [145]. Het factorizatie probleem herleidt zich dan tot de volgende natuurlijke en eenvoudige vraag. Het factorizatie probleem. Neem objecten A en H in C. Beschrijf en klasseer objecten in C die factorizeren door A en H. xviii Het is ook interessant om het omgekeerde probleem te beschouwen; de achterliggende theorie noemen we dubbel gekruiste descent theorie. Onze klassieke inspiratie is descent theorie voor modulen, zie [83]. Onderstel dat A ⊂ E objecten zijn in C. Een factorizatie A-vorm van E is een subobject H van E zodat E factorizeert door A en H. De (mogelijk lege) volledige deelcategorie van C bestaande uit factorizatie A-vormen van E noteren we als F(A, E). Dubbel gekruiste descent theorie bestudeert de volgende vragen. Bestaan van vormen. Neem een uitbreiding A ⊂ E in C. Bestaat er een factorizatie A-vorm van E? Beschrijving en classificatie van vormen. In de onderstelling dat een factorizatie A-vorm van E bestaat, beschrijf en klasseer alle factorizatie A-vormen van E op isomorfisme na. Als we kijken naar het factorizatie probleem voor groepen, dan is een belangrijke stap naar de oplossing van het probleem de constructie van een dubbel gekruist product A ./ H geassocieerd aan een “matched” paar van groepen (A, H, C, B), zoals ingevoerd door Takeuchi [143]. E factorizeert door A en H als en alleen als er een matched paar (A, H, C, B) bestaat zodat E =∼ A ./ H. Het factorizatie probleem kan dus geherformuleerd worden als een zuiver compu- tationeel probleem: Beschrijf de verzameling van alle matched paren (A, H, C, B) en klasseer alle dubbel gekruiste producten op isomorfisme na. In andere categorieen¨ trachten we dezelfde strategie te volgen. Dit werkt goed in het geval van Hopf algebras, voor andere categorieen¨ is dit op dit ogenblik minder duidelijk. We hebben Hoofdstuk 3 als volgt georganizeerd. Voorafgaande begrippen worden besproken in § 3.1; in § 3.2 geven we een aantal zuiver technische resultaten, die van belang zijn voor de rest van het hoofdstuk. In hoofdstelling 3.2.2 geven we een beschrijving van de verzameling van alle Hopf algebra morfismen ψ : A ./ H → A0 ./0 H0 tussen twee dubbel gekruiste producten. In het bijzonder vinden we de verzameling van alle Hopf algebra morfismen tussen semi-directe producten (of smash producten) van Hopf algebras terug in Corollary 3.2.3. In § 3.3 behandelen we het classificatie gedeelte van het factorizatie probleem. Een belangrijke 1 rol is weggelegd voor de groep Hl,c(H,A) bestaande uit luie 1-cocycles van H met coeffici¨ enten¨ in A. hoofdstelling 3.3.7 is de classificatie stelling voor dubbel gekruiste producten: alle Hopf algebras E die factorizeren door een gegeven A en H worden geklasseerd, op isomorfisme dat A stabiliseert na, door een cohomologisch object H2(A, H). Dit wordt geconstrueerd vertrekkende 1 1 van koppels (r, v) ∈ Hl,c(H,A) × Aut CoAlg(H) bestaande uit een lui cocycle en een unitair coalgebra automorfisme v van H, waartussen een bepaalde compatibiliteitsvoorwaarde bestaat. § 3.4 biedt een volledig antwoord tot het derde probleem hierboven, over de classificatie van vormen. We gaan tewerk in vier stappen, die elk hun intrinsiek belang hebben: deformatie van Hopf algebras, deformatie van vormen, beschrijving van vormen, en tenslotte de classificatie van vormen.

In hoofdstelling 3.4.7 geven we een algemene deformatie Hr van een gegeven Hopf algebra H, geassocieerd aan een matched paar van Hopf algebras (A, H, B, C) en een (B, C)-cocycle r : H → A in de zin van Deﬁnition 3.4.4. Hr = H als een coalgebra, en de nieuwe vermenigvuldiging • is gegeven door de formule h • g := h / r(g(1)) g(2), voor alle h, g ∈ Hr = H. Hr is een nieuwe Hopf algebra, genaamd de r-deformatie van H.

xix Neem nu een Hopf algebra uitbreiding A ⊆ E, en de kleine categorie F(A, E) zoals hiervoor beschreven. We noteren het skelet hiervan door F sk(A, E). De factorizatie index [E : A]f is het aantal elementen van F sk(A, E), zie Definition 3.4.2. We noemen de uitbreiding A ⊆ E star als [E : A]f = 1. Het belang van starre extensies wordt ge¨ıllustreerd door de volgende eigenschap: als E/A star is, en A ./ H =∼ A ./0 H0, dan is ook H =∼ H0. Dit is een Krull-Schmidt-Azumaya type eigenschap voor dubbel gekruiste producten: de starre uitbreidingen van Hopf algebras zijn precies diegenen waarvoor de ontbinding als een dubbel gekruist product uniek is. We geven (zeldzame) voorbeelden van starre uitbreidingen, en ook van uitbreidingen met factorizatie index groter dan of gelijk aan 2. Als r : H → A een (B, C)-cocycle is, dan is de r-deformatie Hr r een factorizatie A-vorm van A ./ H, m.a.w. er bestaat een nieuw matched paar (A, Hr, B , C) r ∼ zodanig dat A ./ Hr = A ./ H, zie hoofdstelling 3.4.9. We noemen dit resultaat deformatie van vormen; een andere naam die gebruikt werd voor een gelijkaardig resultaat over algebras is invariantie onder verdraaiingen, zie [76, Theorem 4.4]. De beschrijving van vormen wordt gegeven in hoofdstelling 3.4.10, het omgekeerde resultaat van hoofdstelling 3.4.9: als H een factorizatie A-vorm van E is, dan is elke andere vorm H isomorf met Hr als een Hopf algebra, voor een zeker (B, C)-cocycle r : H → A. Dit resultaat is op zichzelf van belang, omdat het aantoont dat het in wezen volstaat om eén´ object van de categorie F(A, E) te kennen om ze allemaal te kennen. Deze resultaten leiden naar ons hoofdresultaat over de classificatie van vormen, hoofdstelling 3.4.6. Als H een gegeven factorizatie A-vorm van E is, dan bestaat er een bijecie tussen de verzameling van de isomorfisme klassen van factorizerende A-vormen van 2 E en het cohomologische object HA (H,A | (B, C)). Dit leidt ook tot volgende formule voor f 2 de factorizatie index: [E : A] = |HA (H,A | (B, C))|. hoofdstelling 3.4.6 kan beschouwd worden als de dubbel gekruiste versie voor Hopf algebras van de Hilbert 90 stelling: als k ⊆ l een trouw platte uitbreiding van commutatieve ringen is, dan is de relatieve Picard groep Pic(l/k) isomorf met de eerste Amitsur cohomologie groep, zie bijv. [83]. Expliciete voorbeelden worden gegeven in § 3.5: voor twee gegeven Hopf algebras A en H beschrijven we met generatoren en relaties alle Hopf algebras E die factorizeren door A en H. We geven ook de classificatie op isomorfisme na, en bepalen de factorizatie index [E : A]f . Dit verloopt in drie stappen. Eerst berekenen we alle matched paren; dit is het rekentechnische deel, en dit kan niet vermeden worden. Dan beschrijven we de dubbel gekruiste producten met generatoren en relaties. Tenslotte passen we hoofdstelling 3.2.2 toe om de classificatie op isomorfisme na uit te voeren. Als toepassing berekenen we de groep der Hopf algebra automorfismen van een dubbel gekruist product Aut Hopf (A ./ H). Alle resultaten in § 3.4 kunnen vertaald worden naar groepen toe, door de categorie der Hopf algebras te vervangen door die der groepen. Deze resultaten, waarvoor geen bewijs nodig is, worden samengevat in § 3.6.

xx 1 Categorical constructions for Hopf algebras and related topics

1.1 Preliminaries

In this section we introduce the basic framework that will be used throughout this chapter. In what follows k will be a commutative ring. In some specific cases, we will assume that k is a field. Unless specified otherwise, all modules, algebras, coalgebras, bialgebras, tensor products and homomorphisms are over k. Our notation for the standard categories is as follows: kM (k-vector spaces), k-Alg (associative unital k-algebras), k-CoAlg (coalgebras over k), k- BiAlg (bialgebras over k), k-HopfAlg (Hopf algebras over k), MC (right C-comodules), C MD ((C,D)-bicomodules) where C, D are k-coalgebras. For a coalgebra C, we will use Sweedler’s Σ-notation, that is, ∆(c) = c(1) ⊗ c(2), (I ⊗ ∆)∆(c) = c(1) ⊗ c(2) ⊗ c(3), etc. We also use the r Sweedler notation for left and right C-comodules: ρM (m) = m[0] ⊗ m[1] for any m ∈ M if r l l (M, ρM ) is a right C-comodule and ρN (n) = n<−1> ⊗ n<0> for any n ∈ N if (N, ρN ) is a left l C-comodule. Suppose that M is a left C-comodule with structure map ρM : M → C ⊗M and a r r l l r right D-module with structure map ρM : M → M ⊗ D such that (I ⊗ ρM )ρM = (ρM ⊗ I)ρM . We then say that M is a (C,D)-bicomodule . For further details regarding the theory of comodules we refer to [43]. We use the standard notations for opposite and coopposite structures: Aop denotes the opposite of the algebra A and Ccop stands for the coopposite of the coalgebra C. We also require some notions related to homological coalgebra. For basic definitions and properties we refer to [61]. We just recall here, for further reference, the description of the first cohomology group of a coalgebra C with coefficients in a (C,C)-bicomodule N:

0 ∗ l r H (N,C) = {γ ∈ N |(I ⊗ γ)ρN = (γ ⊗ I)ρN } ∗ = {γ ∈ N |n<−1>γ(n<0>) = γ(n[0])n[1], ∀n ∈ N}

Given a vector space V , (K(V ), p) stands for the cofree coalgebra on V , where K(V ) is a coalgebra and p : K(V ) → V is a k-linear map, while (T (V ), i) denotes the tensor algebra, where T (V ) is an algebra and i : V → T (V ) a k-linear map.

1 CHAPTER 1. CATEGORICAL CONSTRUCTIONS

Hopf algebras

Recall that a k-algebra is a k-module together with a multiplication map m = mA : A⊗A → A and a unit map η = ηA : k → A satisfying the conditions: m ◦ (m ⊗ I) = m ◦ (I ⊗ m) m(η ⊗ I) = m(I ⊗ η) = I

Dually, a k-coalgebra C is a k-module together with k-linear maps ∆ = ∆C : C → C ⊗ C and ε = εC : C → k satisfying (∆ ⊗ I) ◦ ∆ = (I ⊗ ∆) ◦ ∆ (ε ⊗ I) ◦ ∆ = (I ⊗ ε) ◦ ∆ = I ∆ is called the comultiplication and ε is called the counit . The second compatibility tells us that the comultiplication is coassociative. Let C be a coalgebra, and A an algebra. Then we can deﬁne a multiplication on Hom(C,A) in the following way: for f, g : C → A, let f ∗ g = mA ◦ (f ⊗ g) ◦ ∆C , that is,

(f ∗ g)(c) = f(c(1))g(c(2)) This multiplication is called the convolution. For a k-module B which is both a k-algebra and a k-coalgebra, the following assertions are equivalent:

1. mH and ηH are comultiplicative;

2. ∆H and εH are multiplicative;

3. for all h, g ∈ H, we have

∆(gh) = g(1)h(1) ⊗ g(2)h(2) ε(gh) = ε(g)ε(h) ∆(1) = 1 ⊗ 1 ε(1) = 1

In this situation, H is called a bialgebra.

A bialgebra H is called a Hopf algebra if the identity IH has an inverse S in the convolution algebra Hom(H,H). Thus, the map S : H → H satisﬁes:

S(h(1))h(2) = h(1)S(h(2)) = η(ε(h)) The map S is called the antipode of H. The antipode of a Hopf algebra is both an antimorphism of algebras as well as an antimorphism of coalgebras.

2 1.1. PRELIMINARIES

Examples 1.1.1 1) Let G be a monoid. Then the group algebra k[G] is a bialgebra with the coalgebra structure given by ∆(g) = g ⊗ g for all g ∈ G. Moreover, k[G] is a Hopf algebra if and only if G is a group. In this case the antipode is given by S(g) = g−1 for all g ∈ G.

2) Recall that a Lie algebra consists of a vector space L together with a linear multiplication [ , ]: L ⊗ L → L such that the following compatibilities hold for all x, y, z ∈ L:

[x, y] = −[y, x]

[x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0 An important class of examples of Lie algebras are those associated to a unitary algebra. If A is an algebra then we can deﬁne on A a Lie multiplication as follows:

[x, y] := xy − yx

The universal enveloping algebra can be constructed as follows: let L be a Lie algebra, (T (L), i) the tensor algebra on the vector space L and deﬁne U(L) := T (L)/I, where I is the ideal of T (L) generated by the elements of the form x ⊗ y − y ⊗ x − [x, y] with x, y ∈ L. U(L) becomes a Hopf algebra with coalgebra structure and antipode given below. The comultiplication and counit are given by the following diagrams:

i i L / U(L) L / U(L) KK D KK D KK DD KK ∆ DD ε f KK g DD K% D! U(L) ⊗ U(L) k where f(x) = x ⊗ 1 + 1 ⊗ x and g(x) = 0 for all x ∈ L. The antipode is given by the following diagram, where p(x) = −x for all x ∈ L:

i L / U(L) EE EE EE S p EE E" U(L)op

3) Let n ≥ 2 and ζ a primitive n-th root of unity. Let Hn2 (ζ) be the algebra deﬁned by the generators c and x with the relations:

cn = 1, xn = 0, xc = ζcx

We can introduce a coalgebra structure on Hn2 (ζ) as follows: ∆(c) = c ⊗ c, ∆(x) = c ⊗ x + x ⊗ 1, ε(c) = 1, ε(x) = 0

Hn2 (ζ) becomes a bialgebra with the above structures. Moreover, Hn2 (ζ) is a Hopf algebra, called the Taft algebra , with the antipode deﬁned as follows: S(c) = cn−1, S(x) = −cn−1x. For n = 2 we obtain Sweedler’s 4-dimensional Hopf algebra (see Section 3.5)

3 CHAPTER 1. CATEGORICAL CONSTRUCTIONS

Basic category theory

Let C be a category and A, B two objects in C. We denote by HomC(A, B) the set of arrows with source A and target B. We will omit the subscript denoting the category, unless confusion is possible. Given a morphism f ∈ C we denote by dom(f) and cod(f) the domain, respectively the codomain of f. A category C is called small if both the objects of C and the arrows of C are sets. If C is a small category we denote by Hom(C) the set of all morphisms of C. We call an arrow f : A → B monomorphism in a category C if for any other object C and for any pair of arrows g, h : C → A, f ◦ g = f ◦ h implies g = h. The dual notion is called epimorphism . A morphism f : A → B is an isomorphism if there exists g : B → A such that f ◦ g = IdB and g ◦ f = IdA. A subobject of an object X is the equivalence class of monomorphisms m : Y → X, where m : Y → X is equivalent to m0 : Y 0 → X if there is an isomorphism p : Y → Y 0 such that m0 ◦ p = m. The category C is called locally small (or well-powered) if the subobjects of each C ∈ C can be indexed by a set. Dually, the category C is colocally small (or co-well-powered) if its dual is locally small. At some point we will also use, in passing, the notion of locally presentable category. More details regarding this type of categories can be found in [2]. If k is a ﬁeld then the categories k-Alg, k-CoAlg, k-BiAlg, k-HopfAlg are locally presentable. An argument for the fact that the category k-HopfAlg is locally presentable is presented in Remark 1.3.2. Let F : I → C be a covariant functor. A limit for F is a pair limF, (pi : limF → F (i))i∈I where limF is an object in C and pi : limF → F (i) are maps in C for all i ∈ I such that:

0 1) for any map γ : i → i in I we have pi0 = F (γ) ◦ pi;

0 2) for any object W ∈ C and any family of maps qi : W → F (i) such that for all γ : i → i in I we have qi0 = F (γ) ◦ qi there exists a unique map q : W → limF in C such that qi = pi ◦ q for every object i ∈ I

A category C is called complete if any functor F : I → C has a limit for all small categories I. We also have the dual notions of colimit respectively cocomplete category. Let F : I → C be a covariant functor. A colimit for F is a pair colimF, (si : F (i) → colimF )i∈I where colimF is an object in C and si : F (i) → olimF are maps in C for all i ∈ I such that:

0 0 1) for any map γ : i → i in I we have si = si ◦ F (γ);

0 2) for any object W ∈ C and any family of maps ti : F (i) → W such that for all γ : i → i in 0 I we have ti = ti ◦ F (γ) there exists a unique map t : colimF → W such that ti = t ◦ si for every object i ∈ I

A category C is called cocomplete if any functor F : I → C has a colimit for all small categories I.

4 1.1. PRELIMINARIES

Examples 1.1.2 1) Take I to be a small discrete category, that is I is a set whose elements are the objects of the category and the morphisms are the identity maps. Then a (co)limit of a functor F : I → C, if it exists, is called a (co)product. More precisely, if I is a small discrete category then a functor F : I → C is essentially nothing but a family of objects (Ci)i∈I in C indexed by the set I. For a further use we recall explicitly the universal properties satisﬁed by the (co)product. Q Q Q A pair i∈I Ci, (πi)i∈I where i∈I Ci is an object in C and pj : i∈I Ci → Cj are maps in C for all j ∈ I is called a product of the family (Ci)i∈I if for each object C ∈ C and each family Q of morphisms fj : C → Cj j∈I there is a unique morphism f : C → i∈I Ci such that the following diagram commutes for all j ∈ I:

C H HH f HH j f HH HH Q H# Ci / Cj i∈I pj

` ` ` A pair i∈I Ci, (qi)i∈I where i∈I Ci is an object in C and qj : Cj → i∈I Ci are maps in C for all j ∈ I is called a coproduct of the family (Ci)i∈I if for each object C ∈ C and each ` family of morphisms fj : Cj → C j∈I there is a unique morphism f : i∈I Ci → C such that the following diagram commutes for all j ∈ I:

qj ` Cj / i∈I Ci HH HH HH f fj HH HH # C

2) Consider I to be a category with two objects A and B and four maps 1A, 1B, r, s, where r, s ∈ Hom(A, B) and define the functor F : I → C as follows: F (A) = X,F (B) = Y,F (r) = f, F (s) = g The (co)limit of functor F defined above is called the (co)equalizer of the pair of maps f, g : X → Y in C. For a further use we recall explicitly the universal properties satisfied by the (co)equalizer. An equalizer of the pair of morphisms f, g : X → Y is a pair (E, p) where E is an object in C and p : E → X is a morphism in C such that for each object E0 ∈ C and each morphism p0 : E0 → X there exists a unique morphism u : E0 → E such that the following diagram commutes: p f E / X / Y O > g / }} u }} }}p0 }} E0 A coequalizer of the pair of morphisms f, g : X → Y is a pair (Q, q) where Q is an object in C and q : Y → Q is a morphism in C such that for each object Q0 ∈ C and each morphism

5 CHAPTER 1. CATEGORICAL CONSTRUCTIONS q0 : Y → Q0 there exists a unique morphism v : Q → Q0 such that the following diagram commutes: f q X / Y / Q g / ? ?? ?? v q0 ?? ? Q0

There are several other important particular examples of (co)limits. However, the (co)products and (co)equalizers seem to be the generic examples as we have the following result:

Theorem 1.1.3 A category C is (co)complete if and only if C has (co)equalizers of all pairs of arrows and all (co)products.

Moreover, we have an explicit way of constructing limits from products and equalizers as follows: Q Let I be a small category, F : I → C be a functor and consider i∈I F (i), (pi)i∈I , Q u∈Hom(I) F (cod(u)), (pu)u∈Hom(I) to be the product in C of the families F (i) i∈I , re- Q Q spectively F (cod(u)) u∈Hom(I) and f, g : i∈I F (i) → u∈Hom(I) F (cod(u)) be the unique maps such that pu ◦ f = pcod(u) and pu ◦ g = F (u) ◦ pdom(u) for all u ∈ Hom(I). Then, the limit of the functor F is the equalizer of the following pair of maps:

Q f Q F (i) / u∈Hom(I) F (cod(u)) i∈I g /

Recall that if (F,G) is a pair of adjoint functors, with F : C → D and G : D → C if and only if there exist two natural transformations η : 1C → GF and ε : FG → 1D, called the unit and the counit of the adjunction, such that G(εD) ◦ ηG(D) = IG(D) and εF (C) ◦ F (ηC ) = IF (C) for all C ∈ C and D ∈ D. A subcategory D of C is called (co)reflective in C when the inclusion functor U : D → C has a (right)left adjoint. Note that Mitchell ([106]) has interchanged the meanings of ”reflection” and ”coreflection”. Let F : C → D be a covariant functor. F generates a universal problem as follows: Given X ∈ D, find an object G(X) ∈ C and a map γ : FG(X)→ X in D such that for each object Y ∈ C and for each map f : F(Y ) → X in D there is a unique map g : Y → G(X) in C such that the following diagram commutes:

F(Y ) (1.1) H HH f F(g) HH HH HH γ H# F G(X) / X

A pair (G(X), γ) that satisﬁes the above conditions is called a universal solution of the universal problem deﬁned by F and X.

6 1.1. PRELIMINARIES

Let G : D → C be a covariant functor. G generates a co-universal problem as follows: Given C ∈ C. Find an object F(C) ∈ D and a map i : C → GFC in C such that for each D ∈ D and for each map f : C → G(D) in C there is a unique map g : F(C) → D in D such that the following diagram commutes:

i / GF(C) (1.2) C G GG GG f GG G(d) GG G# G(D)

A pair (F(C), i) that satisﬁes the above conditions is called a universal solution of the couniversal problem deﬁned by G and C.

Examples 1.1.4 1) Let F1 : k − Alg →k M be the forgetful functor and V be a k-vector space. Then the pair T (V ), i, i : V → T (V ), is a co-universal solution to the co-universal problem generated by F1 and V . Hence, F1 has a left adjoint, the functor associating to a k-vector space its tensor algebra.

2) Let F2 : k − CoAlg →k M be the forgetful functor and V be a k-vector space. Then the pair K(V ), p, p : K(V ) → V is a universal solution to the universal problem generated by F2 and V . Hence, F2 has a right adjoint, the functor associating to a k-vector space its cofree coalgebra.

Theorem 1.1.5 1) Given F : C → D, assume that for each X ∈ D the universal problem deﬁned by F and X has a universal solution. Then there exists a right adjoint functor G : D → C to F. 2) Given G : D → C. Assume that for each C ∈ C the universal problem deﬁned by G and C has a universal solution. Then there is a left adjoint functor F : C → D to G.

r If M is a right C-comodule with structure map ρM and N a left C-comodule with structure map l ρN , the cotensor product MC N is the kernel of the k-linear map: r l ρM ⊗ I − I ⊗ ρN : M ⊗ N → M ⊗ C ⊗ N.

Given comodule maps f : M → M 0 and g : N → N 0, the k-linear map f ⊗ g : M ⊗ N → 0 0 0 0 M ⊗ N induces a k-linear map fC g : MC N → M C N . A left C-comodule M induces C a functor −C M : M → kM. r If ϕ : C → D is a coalgebra map then every right (left) C-comodule (M, ρM ) can be made r r into a right (left) D-comodule with structure map τM : M → M ⊗ D given by τM (m) = ∗ C D m[0] ⊗ ϕ(m[1]). This association deﬁnes a functor ϕC,D : M → M usually called the corestriction functor. C D C D If M ∈ M we obtain a functor −C M : M → M . In particular, C becomes a left D- D C comodule via ϕ and we obtain a functor −DC : M → M called the coinduction functor ∗ which is a right adjoint to the corestriction functor ϕC,D([43, 22.12]).

7 CHAPTER 1. CATEGORICAL CONSTRUCTIONS

We say that a functor F preserves (co)limits if F commutes with (co)limits in the category theoretical sense. For a functor F : C → D we can deﬁne the two functors: op HomC(•, •), HomD(F (•),F (•)) : C × C → Set and a natural transformation given as follows:

F : HomC(•, •) → HomD(F (•),F (•)), FC,C0 (f) = F (f) 0 for any f : C → C in C. F is called a full functor if every FC,C0 is surjective, or, equivalently, has a right inverse. The following categorical results play a key role in showing that the category of Hopf algebras is a coreﬂective subcategory of the category of bialgebras.

Theorem 1.1.6 [93, p. 114] A functor F preserves (co)limits if and only if F preserves (co)products and (co)equalizers.

Theorem 1.1.7 [93, Corollary p. 130] If C is a complete and locally small category with a cogenerator, then a functor G : C → D has a left adjoint if and only if it is limit preserving. Dually, if C is a cocomplete and colocally small category with a generator, then a functor G : C → D has a right adjoint if and only if it is colimit preserving.

Braided monoidal categories and the center construction

Deﬁnition 1.1.8 A monoidal (or tensor) category C = (C, ⊗, I, a, l, r) is a category C endowed with an object I ∈ C called the unit object, a (bi)functor ⊗ : C ×C → C called the tensor product and natural isomorphisms a : ⊗ ◦ (⊗ × C) → ⊗ ◦ (C × ⊗) called the associativity constraint, l : ⊗ ◦ (I × C) → C called the left unit constraint and r : ⊗ ◦ (C × I) → C called the right unit constraint such that the Pentagon Axiom (1.3) and the Triangle Axiom (1.4) are satisﬁed for all objects U, V , W , X of C.

aU,V,W ⊗IdX U ⊗ (V ⊗ W ) ⊗X o (U ⊗ V ) ⊗ W ⊗X (1.3)

aU⊗V,W,X aU,V ⊗W,X (U ⊗ V ) ⊗ (W ⊗ X)

aU,V,W ⊗X IdU ⊗aV,W,X U ⊗ (V ⊗ W ) ⊗ X / U ⊗ V ⊗ (W ⊗ X)

aV,I,W (V ⊗ I) ⊗ W / V ⊗ (I ⊗ W ) (1.4) O OOO ooo OOO ooo rV ⊗IdW OO ooIdV ⊗lW OO' wooo V ⊗ W

8 1.1. PRELIMINARIES

Deﬁnition 1.1.9 (1) Let C = (C, ⊗, I, a, l, r) and D = (D, ⊗, I, a, l, r) be two monoidal categories. A monoidal functor from C to D is a triple (F, ϕ0, ϕ2) where F : C → D is a functor, ϕ0 : I → F (I) is an isomorphism and

ϕ2(U, V ): F (U) ⊗ F (V ) → F (U ⊗ V ) is a family of natural isomorphisms for all objects U, V ∈ C such that the following diagrams commute for all objects U, V , W ∈ C:

aF (U),F (V ),F (W ) F (U) ⊗ F (V )⊗F (W ) / F (U) ⊗ F (V ) ⊗ F (W )

ϕ2(U,V )⊗IdF (W ) IdF (U)⊗ϕ2(V,W ) F (U ⊗ V ) ⊗ F (W ) F (U) ⊗ F (V ⊗ W )

ϕ2(U⊗V,W ) ϕ2(U,V ⊗W )

F (aU,V,W ) F (U ⊗ V ) ⊗ W / F U ⊗ (V ⊗ W )

lF (U) I ⊗ F (U) / F (U) O ϕ0⊗IdF (U) F (lU )

ϕ2(I,U) F (I) ⊗ F (U) / F (I ⊗ U)

rF (U) F (U) ⊗ I / F (U) O IdF (U)⊗ϕ0 F (rU )

ϕ2(U,I) F (U) ⊗ F (I) / F (U ⊗ I)

The monoidal functor (F, ϕ0, ϕ2) is called strict if the isomorphisms ϕ0 and ϕ2 are identities of D. 0 0 0 (2) A natural monoidal transformation η :(F, ϕ0, ϕ2) → (F , ϕ0, ϕ2) between two monoidal functors from C to D is a natural transformation η : F → F 0 such that the following diagrams commute for all objects U, V ∈ C:

η(I) F (I) / F 0(I) aCC {= CC {{ ϕ0 CC {{ϕ0 CC {{ 0 I {

ϕ2(U,V ) F (U) ⊗ F (V ) / F (U ⊗ V )

η(U)⊗η(V ) η(U⊗V ) 0 ϕ2(U,V ) F (U) ⊗ F (V ) / F 0(U ⊗ V )

9 CHAPTER 1. CATEGORICAL CONSTRUCTIONS

A natural monoidal isomorphism is a natural monoidal transformation that is also a natural isomorphism. (3) A monoidal equivalence between monoidal categories is a monoidal functor F : C → D such that there exist a monoidal functor F 0 : D → C and natural monoidal isomorphisms 0 0 η : IdD → FF and θ : F F → IdC.

A monoidal category is called strict whenever the associativity constraint and the unit constraint are the respective identities. Due to a remarkable theorem of MacLane which asserts that any monoidal category is monoidally equivalent to a strict monoidal category, one may always assume that a monoidal category is strict. The categories that we will consider are - technically spoken - not strict, but they can be treated as if they were strict. In a monoidal category one can form n-fold tensor products of an ordered sequence of objects U1, U2, ..., Un in many ways by considering different parenthesizing of the expression U1 ⊗U2 ⊗ ... ⊗ Un, and the products obtained are, in general, distinct objects of C. However, for n = 3, the problem is settled by the associativity isomorphism which gives an isomorphism between the two possible parenthesizings. Moreover, if n = 4 then the pentagon axiom solves this consistency problem by making the two ways to go from U ⊗ (V ⊗ W ) ⊗X to U ⊗ V ⊗ (W ⊗ X) coincide. For n > 5 the problem is solved by the so called Coherence Theorem of MacLane. This allows us to forget about the brackets and freely use the notation U1 ⊗ U2 ⊗ ... ⊗ Un. Let τ : C × C → C × C be the ﬂip functor. A prebraiding on C is a natural transformation c : ⊗ → ⊗ ◦ τ satisfying the following equations, for all U, V, W ∈ C:

cU,V ⊗W = (V ⊗ cU,W ) ◦ (cU,V ⊗ W ); cU⊗V,W = (cU,W ⊗ V ) ◦ (U ⊗ cV,W ). −1 c is called a braiding if it is a natural isomorphism. c is called a symmetry if cU,V = cV,U , for all U, V ∈ C. We refer to [81, XIII.1] for more detail. There is a natural way to associate a (pre)braided monoidal category to a monoidal category. The weak right center Wr(C) of a monoidal category C is the category whose objects are couples of the form (V, c−,V ), with V ∈ C and c−,V : − ⊗ V → V ⊗ − a natural transformation such that c−,I is the natural isomorphism and cX⊗Y,V = (cX,V ⊗ Y ) ◦ (X ⊗ cY,V ), for all X,Y ∈ C. The morphisms are deﬁned in the obvious way. Wr(C) is a prebraided monoidal category; the unit is (I, id), and the tensor product is

0 0 (V, c−,V ) ⊗ (V , c−,V 0 ) = (V ⊗ V , c−,V ⊗V 0 ) where 0 cX,V ⊗V 0 = (V ⊗ cX,V 0 ) ◦ (cX,V ⊗ V ) for all X ∈ C. The prebraiding is given by

0 0 cV,V 0 :(V, c−,V ) ⊗ (V , c−,V 0 ) → (V , c−,V 0 ) ⊗ (V, c−,V ) 0 for all V , V ∈ C. The right center Zr(C) is the full subcategory of Wr(C) consisting of objects (V, c−,V ) with c−,V a natural isomorphism; Zr(C) is a braided monoidal category. For more detail, we refer to [81, XIII.4]. The center construction was independently introduced by Drinfeld (unpublished), Joyal and Street [80] and Majid [98].

10 1.1. PRELIMINARIES

Azumaya algebras

Let k be a commutative ring and A a k-algebra. A(n) will be a shorter notation for the n-fold tensor product A ⊗ · · · ⊗ A. Unadorned ⊗ means ⊗k. AMA is the k-linear category of A- bimodules. An element R ∈ A(3) will be denoted by R = R1 ⊗ R2 ⊗ R3, where summation is implicitly understood. It is well-known that we have a pair of adjoint functors (F,G) between the category of k-modules Mk and the category of A-bimodules AMA. For a k-module N, F (N) = A ⊗ N, with A-bimodule structure a(b ⊗ n)c = abc ⊗ n, for all a, b, c ∈ A and n ∈ N. A ∼ For an A-bimodule M, G(M) = M = {m ∈ M | am = ma, ∀a ∈ A} = AHomA(A, M). The unit η and the counit ε of the adjoint pair (F,G) are given by the formulas

A ηN : N → (A ⊗ N) ; ηN (n) = 1 ⊗ n; A εM : A ⊗ M → M ; εM (a ⊗ m) = am = ma for all n ∈ N, a ∈ A and m ∈ M A. Recall that A is an Azumaya algebra if A is faithfully projective as a k-module, that is, A is ﬁnitely generated, projective and faithful, and the algebra map e op F : A = A ⊗ A → Endk(A),F (a ⊗ b)(x) = axb (1.5) is an isomorphism. Azumaya algebras can be characterized in several ways; perhaps the most natural characterization is the following: A is an Azumaya algebra if and only if the adjoint pair (F,G) is a pair of inverse equivalences, see [83, Theorem III.5.1]. Another characterization is that A is central and separable as a k-algebra.

Separable functors

Recall from [109] that a covariant functor F : C → D is called separable if the natural transformation F : HomC(•, •) → HomD(F (•),F (•)) ; FC,C0 (f) = F (f) splits, that is, there is a natural transformation

P : HomD(F (•),F (•)) → HomC(•, •) such that P ◦ F is the identity natural transformation. Rafael’s Theorem [131] states that the left adjoint F in an adjoint pair of functors (F,G) is separable if and only if the unit of the adjunction η : 1C → GF splits; the right adjoint G is separable if and only if the counit ε : FG → 1D cosplits, that is, there exists a natural transformation ζ : 1D → FG such that ε ◦ ζ is the identity natural transformation. A detailed study of separable functors can be found in [47].

Descent data

(n) Let A be a commutative k-algebra. ⊗ will always mean ⊗k, and A will be a shorter notation for the n-fold tensor product A ⊗ · · · ⊗ A. If V and W are right A-modules, then V ⊗ W is a

11 CHAPTER 1. CATEGORICAL CONSTRUCTIONS

(2) right A -module. Consider a map g : A ⊗ V → V ⊗ A in MA(2) . For a ∈ A and v ∈ V , we P write - temporarily - g(a ⊗ v) = i vi ⊗ ai. Then we have the following three maps in MA(3) P g1 : A ⊗ A ⊗ V → A ⊗ V ⊗ A ; g1(b ⊗ a ⊗ v) = i b ⊗ vi ⊗ ai; P g2 : A ⊗ A ⊗ V → V ⊗ A ⊗ A ; g2(a ⊗ b ⊗ v) = i vi ⊗ b ⊗ ai; (1.6) P g3 : A ⊗ V ⊗ A → V ⊗ A ⊗ A ; g3(a ⊗ v ⊗ b) = i vi ⊗ ai ⊗ b. Let ψ : V ⊗ A → V be the right A-action on V .

Proposition 1.1.10[83, Prop. II.3.1] Assume that g2 = g3 ◦ g1. Then g is an isomorphism if and only if ψ(g(1 ⊗ v)) = v, for all v ∈ V .

In this situation, (V, g) is called a descent datum . A morphism between two descent data (V, g) and (V 0, g0) is a right A-linear map f : V → V 0 such that (f ⊗ A) ◦ g = g0 ◦ (A ⊗ f). The category of descent data is denoted by Desc(A/k). We have a pair of adjoint functors (F,G) between Mk and Desc(A/k). For N ∈ Mk, F (N) = (N ⊗A, g), with g(a⊗n⊗b) = n⊗a⊗b. G(V, g) = {v ∈ V | v ⊗ 1 = g(1 ⊗ v)}. The unit and counit of the adjunction are as follows:

ηN : N → (GF )(N), ηN (n) = n ⊗ 1;

ε(V,g) :(FG)(V, g) = G(V, g) ⊗ A → (V, g), ε(V,g)(v ⊗ a) = va. The Faithfully Flat Descent Theorem can now be stated as follows: if A is faithfully ﬂat over k, then (F,G) is an inverse pair of equivalences. This is essentially [83, Theor´ eme` 3.3], formulated in a categorical language. In [83], a series of applications of descent theory are given, and there exist many more in the literature. Also observe that the descent theory presented in [83] is basically the afﬁne version of Grothendieck’s descent theory [69].

Noncommutative descent theory and comodules over corings

Descent theory can be extended to the case where A are noncommutative. This was done by Cipolla in [54]. After the revival of the theory of corings initiated in [40], it was observed that the results in [54] can be nicely reformulated in terms of corings. Recall that an A-coring C is a coalgebra in the monoidal category of A-bimodules. A right C-comodule is a right A-module M together with a right A-linear map ρ : M → M ⊗A C satisfying appropriate coassociativity and counit conditions. For detail on corings and comodules, we refer to [40, 43]. An important example of an A-coring is Sweedler’s canonical coring C = A ⊗ A. Identifying (A ⊗ A) ⊗A (A ⊗ A) =∼ A(3), we view the comultiplication as a map ∆ : A(2) → A(3). It is given by the formula ∆(a ⊗ b) = a ⊗ 1 ⊗ b. The counit ε is given by ε(a ⊗ b) = ab. For a right A-module M, ∼ we can identify M ⊗A (A ⊗ A) = M ⊗ A. A right A ⊗ A-comodule is then a right A-module V together with a right k-linear map ρ : V → V ⊗ A, notation ρ(v) = v[0] ⊗ v[1] satisfying the relations

v[0]v[1] = v; (1.7)

ρ(v[0]) ⊗ v[1] = v[0] ⊗ 1 ⊗ v[1]; (1.8)

ρ(va) = v[0] ⊗ v[1]a (1.9)

12 1.1. PRELIMINARIES for all v ∈ V and a ∈ A. The category of right A ⊗ A-comodules is denoted by MA⊗A. There A⊗A is an adjunction between Mk and M . Cipolla’s descent data are nothing else then A ⊗ A- comodules, and Cipolla’s version of the Faithfully Flat Descent Theorem asserts that this is a pair of inverse equivalences if A is faithfully ﬂat over k, we refer to [46] for a detailed discussion. First observe that this machinery works for a general extension k → A of rings, that is, A and k are not necessarily commutative. In this note, however, we keep k commutative. If A is commutative, then the categories Desc(A/k) and MA⊗A are isomorphic. (V, g) ∈ Desc(A/k) corresponds to (V, ρ) ∈ MA⊗A, with ρ(v) = g(1 ⊗ v). Sometimes it is argued that this generalization is not satisfactory, since there is no counterpart to Proposition 1.1.10 in the case where A is noncommutative. In Section 1.6 we will present an appropriate generalization Desc(A/k) to the noncommutative situation, with a suitable generalized version of Proposition 1.1.10, see Proposition 1.6.4 and Remark 1.6.8.

13 CHAPTER 1. CATEGORICAL CONSTRUCTIONS

1.2 Categorical constructions for Hopf algebras

Throughout this section, k will be a field. It is well known that the category k-Alg of k-algebras is complete and cocomplete. This is immediately implied by the existence of products, coproducts, equalizers and coequalizers in the category k-Alg. The categories of coalgebras and bialgebras have arbitrary coproducts and coequalizers (see [58, Propositon 1.4.19], [43, Proposi- tion 2.10], [118, Corollary 2.6.6] hence these categories are cocomplete. In this section we prove that the category of Hopf algebras has coproducts and coequalizers. Related to the question of whether these categories are complete (i.e. if they have arbitrary products and equalizers) we could not find similar results in the classical Hopf algebra textbooks ([1], [139]), not even in the more recent ones ([43], [58], [81], [106]). For example, [58, Propositon 1.4.21] proves only the existence of finite products (namely the tensor product of coalgebras) and only in the category of cocommutative coalgebras, as a dual result to the one concerning commutative algebras. We shall fill this gap: using the fact that the forgetful functor from the category of coalgebras to the category of vector spaces has a right adjoint, namely the so called cofree coalgebra, we shall construct explicitly the product of an arbitrary family of coalgebras. As a consequence, the product of an arbitrary family of bialgebras and Hopf algebras is constructed. The equalizers of two morphisms of coalgebras (bialgebras, Hopf algebras) are also described explicitly. Thus we shall obtain that the categories of coalgebras, bialgebras and Hopf algebras are complete and a complete description for limits in the above categories is given. We start by constructing coequalizers and coproducts in the category of Hopf algebras. Let (A, mA, ηA, ∆A, εA), (B, mB, ηB, ∆B, εB) be two bialgebras and f, g : B → A be two bialgebra maps. Consider I the two-sided ideal generated by {f(b) − g(b) | b ∈ B }. By a simple computation it can be seen than I is also a coideal. Then (A/I, π) is the coequalizer of the morphisms (f, g) in k-BiAlg, where π : A → A/I is the canonical projection. Indeed for all bialgebras H and all bialgebra morphisms h : A → H such that h ◦ f = h ◦ g we obtain I ⊆ kerh, hence there exists an unique bialgebra map h0 : A/I → H such that h0 ◦ π = h.

Remark 1.2.1 Note that if A, B are two Hopf algebras and f, g : B → A are Hopf algebra maps then the ideal I deﬁned above is actually a Hopf ideal and (A/I, π) is the coequalizer of the morphisms (f, g) in k-HopfAlg, where π : A → A/I is the canonical projection.

Next, we recall from [118] the construction of the coproduct in the category k-BiAlg of bial- L gebras. Let (Al)l∈I be a family of algebras, l∈I Al, (jl)l∈I be the coproduct in kM and L L L i : l∈I Al → T l∈I Al be the canonical inclusion where T l∈I Al is the tensor alge- L ` L bra of the vector space l∈I Al. Then l∈I Al := T l∈I Al /L, (ql)l∈I is the coprod- L uct of the above family in k-Alg, where L is the two sided ideal in T Al generated by l∈I the set J := {i ◦ jl(xlyl) − i jl(xl) i jl(yl) , 1 L − i ◦ jl(1Al )|xl, yl ∈ Al, l ∈ I}, T Al L L ν : T l∈I Al → T l∈I Al /L denotes the canonical projection and ql = ν ◦ i ◦ jl for all ` l ∈ I. Furthermore, l∈I Al is actually a bialgebra provided that (Al)l∈I is a family of bialgebras. The comultiplication and the counit are given by the unique algebra maps such that the

14 1.2. CATEGORICAL CONSTRUCTIONS FOR HOPF ALGEBRAS following diagrams commute:

ql ` ql ` Al / l∈I Al Al / l∈I Al (1.10) NNN GG NNN GG NN ∆ GG ε NN εl G (ql⊗ql)◦∆l NN GG ` & ` G# l∈I Al ⊗ l∈I A k

` and l∈I Al is the coproduct in k-BiAlg of the above family of bialgebras ([118, Corollary 2.6.2]). ` Now let Hl, ml, ηl, ∆l, εl,Sl l∈I be a family of Hopf algebras. Consider (H := l∈I Hl, m, η, ∆, ε), (ql)l∈I the coproduct of the above family in the category k-BiAlg of bialgebras. The universal property of the coproduct yields an unique bialgebra map S : H → Hopcop such that the following diagram commutes for all l ∈ I:

ql Hl / H (1.11)

Sl S q opcop l opcop Hl / H

With the notations above we have the following result which provides a completely description of the coproducts in the category k-HopfAlg of Hopf algebras:

Theorem 1.2.2 Let Hl, ml, ηl, ∆l, εl,Sl l∈I be a family of Hopf algebras. The Hopf algebra ` H := Hl, m, η, ∆, ε, S together with structure maps (ql)l∈I is the coproduct in the l∈I category k-HopfAlg of the family Hl, ml, ηl, ∆l, εl,Sl l∈I of Hopf algebras. In particular, the category k-HopfAlg is cocomplete.

Proof: We will ﬁrst prove that S is an antipode for the bialgebra H, i.e.

m ◦ Id ⊗ S◦∆ = m ◦ S ⊗ Id◦∆ = η ◦ ε (1.12)

Since S : H → Hopcop deﬁned in (1.11) is a bialgebra map we only need to prove that (1.12) holds only on the generators of H as an algebra. Indeed, let h, k be generators in H for which (1.12) holds. We obtain :

(hk)(1)S((hk)(2)) = h(1)k(1)S(k(2))S(h(2)) = ε(k)h(1)S(h(2)) = ε(h)ε(k)1H = ε(hk)1H

It follows from here that (1.12) also holds for kh and thus it holds for all elements in H. L Now having in mind that H := T l∈I Hl /L we only need to prove (1.12) for the elements L L x ∈ H with x ∈ l∈I Hl, whereas the tensor algebra T l∈I Hl is the free algebra on Lb L Q l∈I Hl. Moreover, since l∈I Hl = {x ∈ l∈I Hl | supp (x) < ∞} it is enough to show

15 CHAPTER 1. CATEGORICAL CONSTRUCTIONS

that (1.12) holds for all xl ∈ Hl, l ∈ I. We then have: m ◦ Id ⊗ S ◦∆(xbl) = m ◦ Id ⊗ S ◦∆ ◦ ql(xl) (1.10) = m ◦ Id ⊗ S ◦(ql ⊗ ql) ◦ ∆l(xl) = m ◦ ql ⊗ (S ◦ ql) ◦∆l(xl) (1.11) = m ◦ ql ⊗ (ql ◦ Sl) ◦∆l(xl)

= m ◦ (ql ⊗ ql) ◦ (Id ◦ Sl) ◦ ∆l(xl)

ql− algebra map = ql ◦ ml ◦ (Id ◦ Sl) ◦ ∆l(xl)

= ql ◦ ηl ◦ εl(xl)

ql− algebra map = η ◦ εl(xl)

ql− coalgebra map = η ◦ ε ◦ ql(xl) = η ◦ ε(xbl) Hence m ◦ Id ⊗ S◦∆ = η ◦ ε. In the same way it can be proved that m ◦ S ⊗ Id◦∆ = η ◦ ε. Thus S is an antipode for H, as desired. Now since k-HopfAlg is a full subcategory of the category k-BiAlg it follows that (H := ` l∈I Hl, m, η, ∆, ε), (ql)l∈I is also the coproduct of the family Hl, ml, ηl, ∆l, εl,Sl l∈I of Hopf algebras in the category k-HopfAlg.

First, we explicitly construct the product of an arbitrary family of coalgebras.

Theorem 1.2.3 The category k-CoAlg of coalgebras has arbitrary products and equalizers. In particular, the category k-CoAlg of coalgebras is complete.

Proof: Let f, g : C → D be two coalgebra maps and S := {c ∈ C | f(c) = g(c) }, which is a k-subspace of C. Let E be the sum of all subcoalgebras of C included in S. Note that the family of subcoalgebras of C included in S is not empty since it contains the null coalgebra. It is immediate that E is a subcoalgebra of C. We shall prove that (E, i) is the equalizer of the pair (f, g) in k-CoAlg, where i : E → C is the canonical inclusion. Let E0 be a coalgebra and h : E0 → C a coalgebra map such that f ◦ h = g ◦ h. Then fh(x)= gh(x), for all x ∈ E0, hence h(E0) ∈ S and since h(E0) is a subcoalgebra in C we obtain h(E0) ⊆ E. Thus there exists a unique coalgebra map h : E0 → E such that i ◦ h = h. Hence (E, i) is the equalizer of the pair (f, g) in the category k-CoAlg of coalgebras. Q Now let (Ci)i∈I be a family of coalgebras and i∈I Ci, (πi)i∈I be the product of the k- Q Q modules (Ci)i∈I . Let K( i∈I Ci), p be the cofree coalgebra over the vector space i∈I Ci. Q Let D be the sum of all subcoalgebras E of K i∈I Ci such that πi ◦ p ◦ jE is a coalgebra Q map for all i ∈ I, where jE : E → K( i∈I Ci) is the canonical inclusion. The family of subcoalgebras of E satisfying this property is nonempty since it contains the null coalgebra. The Q k-linear map πi ◦ p ◦ j : D → Ci is a coalgebra map for all i ∈ I where j : D → K( i∈I Ci)

16 1.2. CATEGORICAL CONSTRUCTIONS FOR HOPF ALGEBRAS

is the canonical inclusion. We shall prove that D, (πi ◦ p ◦ j)i∈I is the product of the family of coalgebras (Ci)i∈I in k-CoAlg. 0 0 Indeed, let D be a coalgebra and gi : D → Ci, i ∈ I, a family of coalgebra maps. Using the universal property of the product in kM we obtain that there exists a unique k−linear map 0 Q Q g : D → i∈I Ci such that πi ◦ g = gi, for all i ∈ I. Furthermore, since K( i∈I Ci), p is Q 0 the cofree coalgebra over the k−module i∈I Ci, there exists a unique coalgebra map f : D → Q K( i∈I Ci) such that p ◦ f = g. Thus we have the following commutative diagram:

D0 X VKVXVXVXXX KK VVVXVXXXX KK VVV XXXXX gi KK VVVV XXXXX KK g VVVV XXXXX f K% VVVV XXXXX Q V* Q XXXXX D / K Ci / Ci X/, Ci j i∈I p i∈I πi

0 It follows that (πi ◦ p)(f(x)) = gi(x), for all x ∈ D and i ∈ I. So, since each gi is a coalgebra map, we have f(D0) ⊆ D. Hence, we proved that for any coalgebra D0 and any 0 0 family gi : D → Ci, i ∈ I, of coalgebra maps there exists a coalgebra map f : D → D such 0 that (πi ◦ p ◦ j) ◦ f = gi, for all i ∈ I. Let h : D → D be another coalgebra map such that (πi ◦ p ◦ j) ◦ h = gi for all i ∈ I. From the uniqueness of g we obtain p ◦ j ◦ h = g. Moreover, from the uniqueness of f we obtain j ◦ h = f, hence h = f. Thus D, (πi ◦ p ◦ j)i∈I is the product of the family (Ci)i∈I in the category k-CoAlg of coalgebras.

Remark 1.2.4 1) In [21, Lemma 1.1.3] a description for the equalizers in the category k- HopfAlg is given. We can use the same method in order to obtain another description for the equalizer of a pair of coalgebra (or bialgebra) maps. Let f, g : C → D be two coalgebra maps. It can be easily proved that (E, i) is the equalizer of the pair (f, g) in the category k-CoAlg of coalgebras, where

E = {c ∈ C | c(1) ⊗ f(c(2)) ⊗ c(3) = c(1) ⊗ g(c(2)) ⊗ c(3) } and i : E → C is the canonical inclusion. This equivalent description of equalizers in the category k-CoAlg will turn out to be more efﬁcient for computations.

Example 1.2.5 Let G be a multiplicative group and kG the k-vector space with basis {g|g ∈ G} endowed with the classical coalgebra structure : ∆(g) = g ⊗ g and ε(g) = 1 for all g ∈ G. P Thus any element x ∈ kG has the form x = g∈G kgg where (kg)g∈G is a family of elements in k with only a ﬁnite number of non-zero elements. We use the following formal notation x−1 := P −1 −1 g∈G kgg and 0 = 0. −1 Consider the coalgebra maps f = IdkG and h : kG → kG given by h(g) = g for all g ∈ G. Then, in the light of the above remark, it follows that the equalizer of the morphisms (f, g) is given by the pair (E, i) where E = {x ∈ kG|x ⊗ x ⊗ x = x ⊗ x−1 ⊗ x} and i is the canonical inclusion.

As an easy consequence of [93, Chapter 5 §2, Theorem 1] we obtain the following description for limits in the category k-CoAlg of coalgebras:

17 CHAPTER 1. CATEGORICAL CONSTRUCTIONS

Remark 1.2.6 Let J be a small category, F : J →k-CoAlg be a functor, Πj∈J F (j), (pj)j∈J , Πu∈Hom(J)F (cod(u)), (pu)u∈Hom(J) be the product in k-CoAlg of the families F (j) j∈J , respectively F (cod(u)) u∈Hom(J) and f, g :Πj∈J F (j) → Πu∈Hom(J)F (cod(u)) be the unique coalgebra maps such that pu ◦ f = pcod(u) and pu ◦ g = F (u) ◦ pdom(u) for all u ∈ Hom(J). We deﬁne

D = {x ∈ Πj∈J F (j)|x(1) ⊗ f(x(2)) ⊗ x(3) = x(1) ⊗ g(x(2)) ⊗ x(3)}. Then the pair D, (ϕj = pj ◦ e)j∈J is the limit of the functor F , where e : D → Πj∈J F (j) is the canonical inclusion.

In what follows we will make use of Theorem 1.2.3 in order to construct the product in the category of k-BiAlg of bialgebras.

Theorem 1.2.7 The category k-BiAlg of bialgebras has arbitrary products and equalizers. In particular, the category k-BiAlg of bialgebras is complete.

Q Proof: Let Bi, mi, ηi, ∆i, εi i∈I be a family of bialgebras and ( i∈I Bi, ∆, ε), (πi)i∈I be the product of this family in the category k-CoAlg of coalgebras. Since (Bi, mi, ηi, ∆i, εi) is a bialgebra it follows that mi : Bi ⊗ Bi → Bi and ηi : k → Bi are coalgebra maps for all i ∈ I. Q Then there exists a unique coalgebra map η : k → i∈I Bi such that the following diagram :

k (1.13) HH HH ηi η HH HH HH Q πi # i∈I Bi / Bi Q Q is commutative for all i ∈ I. Also there exists a unique coalgebra map m : i∈I Bi⊗ i∈I Bi → Q i∈I Bi for which the diagram : Q Q i∈I Bi ⊗ i∈I Bi (1.14) N NNmi◦(πi⊗πi) m NNN NNN NNN Q πi N' i∈I Bi / Bi is commutative for all i ∈ I. Q First, we will prove that ( i∈I Bi, m, η) is a k-algebra. Since πi ◦ m ◦ (m ⊗ Id) is a coalgebra map by the universal property of the product we obtain that there exists a unique coalgebra map Q ⊗3 Q ψ : i∈I Bi → i∈I Bi such that the following diagram:

Q ⊗3 i∈I Bi J JJπi◦m◦(m⊗Id) ψ JJ JJ JJ Q πi J$ i∈I Bi / Bi

18 1.2. CATEGORICAL CONSTRUCTIONS FOR HOPF ALGEBRAS is commutative for all i ∈ I. It is easy to see that the coalgebra map m ◦ (m ⊗ Id) makes the above diagram commute. Thus, using the uniqueness of ψ, in order to prove that m◦(m⊗Id) = m ◦ (Id ⊗ m) it is enough to show that πi ◦ m ◦ (m ⊗ Id) = πi ◦ m ◦ (Id ⊗ m) for all i ∈ I. We have :

(1.14) πi ◦ m ◦ (m ⊗ Id) = mi ◦ (πi ⊗ πi) ◦ (m ⊗ Id) = mi ◦ (πi ◦ m) ⊗ πi (1.14) = mi ◦ mi ◦ (πi ⊗ πi) ⊗πi

= mi ◦ (mi ⊗ Id) ◦ (πi ⊗ πi ⊗ πi)

= mi ◦ (Id ⊗ mi) ◦ (πi ⊗ πi ⊗ πi) = mi ◦ πi ⊗ mi ◦ (πi ⊗ πi) (1.14) = mi ◦ πi ⊗ (πi ◦ m)

= mi ◦ (πi ⊗ πi) ◦ (Id ⊗ m) (1.14) = πi ◦ m ◦ (Id ⊗ m)

Hence m ◦ (m ⊗ Id) = m ◦ (Id ⊗ m), i.e. m is associative.

Consider now the coalgebra map πi ◦ m ◦ (η ⊗ Id). From the universal property of the product, Q we obtain that there exists a unique coalgebra map ϕ : k⊗ i∈I Bi → Bi such that the following diagram : Q k ⊗ i∈I Bi J JJπi◦m◦(η⊗Id) ϕ JJ JJ JJ Q πi J% i∈I Bi / Bi is commutative for all i ∈ I. By the argument above, in order to prove that m ◦ (η ⊗ Id) = s it will be enough to show that πi ◦ m ◦ (η ⊗ Id) = πi ◦ s, where we denote by s the scalar multiplication. We have:

(1.14) πi ◦ m ◦ (η ⊗ Id) = mi ◦ (πi ⊗ πi) ◦ (η ⊗ Id) = mi ◦ (πi ◦ η) ⊗ πi (1.13) = mi ◦ (ηi ⊗ πi)

= mi ◦ (ηi ⊗ Id) ◦ (Id ⊗ πi)

= s ◦ (Id ⊗ πi) Q Furthermore, let k1 ⊗ b ∈ k ⊗ i∈I Bi. Having in mind that πi is a k-linear map we obtain :

s ◦ (Id ⊗ πi)(k1 ⊗ b) = k1πi(b)

= πi(k1b)

= πi ◦ s(k1 ⊗ b)

19 CHAPTER 1. CATEGORICAL CONSTRUCTIONS

Thus we proved that πi ◦m◦(η ⊗Id) = πi ◦s. In the same way it follows that m◦(Id⊗η) = s. Q Hence ( i∈I Bi, m, η) is an algebra and since m and η are coalgebra maps we obtain that Q ( i∈I Bi, m, η, ∆, ε) is actually a bialgebra. Q To end the proof we still need to show that ( Bi, m, η, ∆, ε) is the product of the family i∈I Bi, mi, ηi, ∆i, εi i∈I in the category k-BiAlg. Let (B, mB, ηB, ∆B, εB) be a bialgebra and (gi)i∈I be a family of bialgebra maps, gi : B → Bi for all i ∈ I. From the universal property of Q the product, we obtain that there exists an unique coalgebra map θ : B → i∈I Bi such that the following diagram commutes :

B (1.15) HH HH gi θ HH HH HH Q πi # i∈I Bi / Bi

We only need to prove that θ is also an algebra map. By the argument used above, it is enough to show that:

πi ◦ θ ◦ mB = πi ◦ m ◦ (θ ⊗ θ) and πi ◦ θ ◦ ηB = πi ◦ η (1.16)

Having in mind that gi is an algebra map, we have:

(1.14) πi ◦ m ◦ (θ ⊗ θ) = mi ◦ (πi ⊗ πi) ◦ (θ ⊗ θ) = mi ◦ (πi ◦ θ) ⊗ (πi ◦ θ) (1.15) = mi ◦ (gi ⊗ gi)

= gi ◦ mB (1.15) = πi ◦ θ ◦ mB

(1.15) (1.13) Moreover, πi ◦ θ ◦ ηB = gi ◦ ηB = ηi = πi ◦ η hence (1.16) holds.

In what follows we construct equalizers. Let (A, mA, ηA, ∆A, εA), (B, mB, ηB, ∆B, εB) be two bialgebras and f, g : B → A be two bialgebra maps. We denote by S := {b ∈ B|f(b) = g(b)}. Let D be the sum of all subcoalgebras of B contained in S. We already noticed before that the family of subcoalgebras of B with this property is nonempty and that D is a subcoalgebra of B. The pair (D, i) is the equalizer of the morphisms (f, g) in k-BiAlg, where i : D → B is the canonical inclusion. We only need to prove that D is actually a subbialgebra of B. Consider Pn q = k=1 dik ⊗ djk ∈ D ⊗ D. We then have:

n X mA ◦ (f ⊗ f)(q) = mA f(dik ) ⊗ f(djk ) k=1 n X = mA g(dik ) ⊗ g(djk ) k=1 = mA ◦ (g ⊗ g)(q)

20 1.2. CATEGORICAL CONSTRUCTIONS FOR HOPF ALGEBRAS

Now having in mind that f and g are algebra maps we obtain f mB(D⊗D) = g mB(D⊗D) , hence mB(D⊗D) ⊆ S and since mB(D⊗D) is a subcoalgebra it follows that mB(D⊗D) ⊆ D. Thus D is a subbialgebra of B and it can be shown as in Theorem 1.2.3 that the pair (D, i) is the equalizer of the morphisms (f, g) in k-BiAlg.

As remarked before, we can obtain a description for the equalizers in k-BiAlg similar to the one in Remark 1.2.4. Thus, we have the following description for limits in k-BiAlg: Remark 1.2.8 Let J be a small category, F : J →k-BiAlg be a functor, Πj∈J F (j), (pj)j∈J , Πu∈Hom(J)F (cod(u)), (pu)u∈Hom(J) be the product in k-BiAlg of the families F (j) j∈J , respectively F (cod(u)) u∈Hom(J) and f, g :Πj∈J F (j) → Πu∈Hom(J)F (cod(u)) be the unique bialgebra maps such that pu ◦ f = pcod(u) and pu ◦ g = F (u) ◦ pdom(u) for all u ∈ Hom(J). We deﬁne

Theorem 1.2.9 The category k-HopfAlg of Hopf algebras has arbitrary products and equalizers. In particular, the category k-HopfAlg of Hopf algebras is complete.

Q Proof: Let Hi, mi, ηi, ∆i, εi,Si i∈I be a family of Hopf algebras and (B := i∈I Hi, ∆, ε, m, η), (πi)i∈I be the product of this family in the category k-BiAlg of bialgebras. The universal property of the product yields an unique bialgebra map S : Bop,cop → B such that the following diagram commutes for all i ∈ I:

πi B / Bi (1.17) O O S Si

op,cop πi op,cop B / Bi

Let H be the sum of all subcoalgebras C of the bialgebra B such that : S(c(1))c(2) = c(1)S(c(2)) = η ◦ ε(c) for all c ∈ C. The family of subcoalgebras C satisfying the above property is nonempty by the same argument used in the proof of Theorem 1.2.3. Moreover, it is easy to see that

S(h(1))h(2) = h(1)S(h(2)) = η ◦ ε(h) (1.18) for all h ∈ H. We will prove that H is a bialgebra and it will follow by (1.18) that H is actually a Hopf algebra with the antipode S|H . First note that η(k) = k1B ⊆ H. Let h, g ∈ H. We then have: S (hg)(1) (hg)(2) = S(h(1)g(1))h(2)g(2)

= S(g(1))S(h(1))h(2)g(2)

= S(g(1))(η ◦ ε)(h)g(2) = η ◦ ε(h)η ◦ ε(g)= η ◦ ε(hg)

21 CHAPTER 1. CATEGORICAL CONSTRUCTIONS

In the same way it can be proved that (hg)(1)S (hg)(2) = η ◦ ε(hg). Thus hg ∈ H and H is indeed a bialgebra. In order to conclude that S|H is an antipode for H we need to prove that S(H) ⊆ H. Let h ∈ H ; we obtain: S S(h)(1) S(h)(2) = S S(h(2)) S(h(1)) = S h(1)S(h(2)) = Sη ◦ ε(h) = η ◦ ε(h) = η ◦ εS(h) A similar computation shows that we also have S(h)(1)S S(h)(2) = η ◦ ε S(h) for all h ∈ H. Hence H is a Hopf algebra with S|H as antipode. To end the proof we still need to show that (H, m, η, ∆, ε, S|H ) , (qi)i∈I is the product of the family Hi, mi, ηi, ∆i, εi,Si i∈I in the category k-HopfAlg, where qi := πi ◦ j for all i ∈ I and j : H → B is the canonical inclusion. Let K be a Hopf algebra with antipode SK and fi : K → Hi be a family of Hopf algebra maps for all i ∈ I. Since B is the product in k-BiAlg of the above family of Hopf algebras, there exist a unique morphism of bialgebras f : K → B such that the following diagram commutes:

K (1.19) AA AA fi f AA AA πi B / Hi

Using the fact that fi is a Hopf algebra map we have :

(1.17) πi ◦ S ◦ f = Si ◦ πi ◦ f (1.19) = Si ◦ fi

= fi ◦ SK (1.19) = πi ◦ f ◦ SK for all i ∈ I. By the same argument used in the proof of theorem Theorem 1.2.7 it follows that:

S ◦ f = f ◦ SK (1.20)

Thus, for all k ∈ K we have:

S(f(k)(1))f(k)(2) = S(f(k(1)))f(k(2)) (1.20) = f SK (k(1)) f(k(2)) = f SK (k(1))k(2) = f(k)

Hence f(K) ⊆ H. Thus, we obtained an unique Hopf algebra map f : K → H such that qi ◦ f = fi for all i ∈ I. Now, since the forgetful functor U : k-HopfAlg → k-BiAlg has a left

22 1.2. CATEGORICAL CONSTRUCTIONS FOR HOPF ALGEBRAS adjoint (see [142]) it follows that, in particular, U preserves products. That is, H = B and the map S obtained in (1.17) is actually an antipod for B. Thus, (B, m, η, ∆, ε, S) , (πi)i∈I is the product of the family Hi, mi, ηi, ∆i, εi,Si i∈I in the category k-HopfAlg. Now let f, g :H → K be two Hopf algebra morphisms and S := {h ∈ H|f(h) = g(h)}, which is just a k-subspace of H. Let D be the sum of all subcoalgebras of H contained in S. Again, the family of subcoalgebras of H included in S is not empty by the same argument used in Theorem 1.2.3. A simple computation shows that D is in fact a Hopf subalgebra of H. Moreover, (D, i) is the equalizer in the category k-HopfAlg of the pair (f, g) where i : D → H is the canonical inclusion.

Remark 1.2.10 Let J be a small category, F : J →k-HopfAlg be a functor, Πj∈J F (j), (pj)j∈J , Πu∈Hom(J)F (cod(u)), (pu)u∈Hom(J) be the product in k-HopfAlg of the families F (j) j∈J , respectively F (cod(u)) u∈Hom(J) and f, g :Πj∈J F (j) → Πu∈Hom(J)F (cod(u)) be the unique Hopf algebra maps such that pu ◦ f = pcod(u) and pu ◦ g = F (u) ◦ pdom(u) for all u ∈ Hom(J). We deﬁne

23 CHAPTER 1. CATEGORICAL CONSTRUCTIONS

1.3 Reﬂective and coreﬂective subcategories

Recall that the forgetful functor from the category of groups to the category of monoids has a left adjoint, the so-called anvelopant group of a monoid, and a right adjoint, which assigns to each monoid the group of its invertible elements. Therefore, if we think of Hopf algebras as a natural generalization of groups, we may expect the same behavior in the case of the forgetful functor F : k-HopfAlg → k-BiAlg from the category of Hopf algebras to the category of bialgebras. It is well known that the above forgetful functor F has a left adjoint [118, Theorem 2.6.3]. We shall prove in this section that F has also a right adjoint. Thus, we can conclude that the category of Hopf algebras is a reflective and coreflective subcategory of the category of bialgebras. Furthermore, in [139, p. 62] it is shown that the category of bialgebras is a reflective subcategory of the category of coalgebras. Moreover, it is stated that the category of bialgebras is also a coreflective subcategory of the category of algebras ([139, p. 134]). In this section we give an affirmative answer to Sweedler’s last statement. Putting all together, the existence of a right adjoint of the forgetful functor from the category of Hopf algebas to the category of algebras is proven.

Theorem 1.3.1 The forgetful functor F : k − BiAlg → k − Alg has a right adjoint, i.e. the category k-BiAlg is a coreﬂective subcategory of k-Alg.

Proof: Let (A, m, η) be an algebra. In order to prove that F has right adjoint it is enough to show that the couniversal problem generated by the algebra A and the functor F has a couniversal solution [119, Proposition 5.22]. Let (C(A), ∆, ε), p be the cofree coalgebra over A. From the universal property of the cofree coalgebra we obtain that there exists a unique coalgebra map η : C(A) ⊗ C(A) → C(A) such that the following diagram is commutative:

p C(A) / A (1.21) s9 O ss ss m ss ssm◦(p⊗p) ss C(A) ⊗ C(A)

By the same argument, there exists a unique coalgebra map η : k → C(A) such that the following diagram is commutative: p C(A) / A (1.22) O z< zz η zz zz η zz C

First we will prove that C(A), ∆, ε, m, η is a bialgebra. Since p ◦ m ◦ Id ⊗ m is a k-linear

24 1.3. REFLECTIVE AND COREFLECTIVE SUBCATEGORIES map there exists a unique coalgebra map ψ : C(A)⊗3 → C(A) such that the following diagram:

p C(A) / A O x< xx ψ xx xxp◦m◦(Id⊗m) xx C(A)⊗3 is commutative. It is easy to see that the coalgebra map m ◦ (Id ⊗ m) makes the above diagram commute. Thus, using the uniqueness of ψ, in order to prove that m ◦ (m ⊗ Id) = m ◦ (Id ⊗ m) it is enough to show that p ◦ m ◦ (m ⊗ Id) = p ◦ m ◦ (Id ⊗ m). We have :

(1.21) p ◦ m ◦ (m ⊗ Id) = m ◦ (p ⊗ p) ◦ (m ⊗ Id) = m ◦ (p ◦ m) ⊗ p (1.21) = m ◦ m ◦ (p ⊗ p)⊗p = m ◦ (m ⊗ Id) ◦ (p ⊗ p ⊗ p) = m ◦ (Id ⊗ m) ◦ (p ⊗ p ⊗ p) = m ◦ p ⊗ m ◦ (p ⊗ p) (1.21) = m ◦ p ⊗ (p ◦ m) = m ◦ (p ⊗ p) ◦ (Id ⊗ m) (1.21) = p ◦ m ◦ (Id ⊗ m)

Hence m ◦ (m ⊗ Id) = m ◦ (Id ⊗ m). Consider now the k-linear map p ◦ m ◦ (η ⊗ Id). From the universal property of the cofree coalgebra, we obtain that there exists a unique coalgebra map ϕ : k ⊗ C(A) → C(A) such that the following diagram : p C(A) / ; A O vv vv ϕ vv vvp◦m◦(η⊗Id) vv k ⊗ C(A) is commutative. By the argument above, in order to prove that m◦(η ⊗Id) = s it will be enough to show that p ◦ m ◦ (η ⊗ Id) = p ◦ s, where we denoted by s the scalar multiplication. We have:

(1.21) p ◦ m ◦ (η ⊗ Id) = m ◦ (p ⊗ p) ◦ (η ⊗ Id) = m ◦ (p ◦ η) ⊗ p (1.22) = m ◦ (η ⊗ p) = m ◦ (η ⊗ Id) ◦ (Id ⊗ p) = s ◦ (Id ⊗ p)

25 CHAPTER 1. CATEGORICAL CONSTRUCTIONS

Furthermore, let k1 ⊗ a ∈ k ⊗ C(A). Having in mind that p is a k-linear map we obtain :

s ◦ (Id ⊗ p)(k1 ⊗ a) = k1p(a)

= p(k1a)

= p ◦ s(k1 ⊗ a)

Hence we proved that C(A), ∆, ε, m, η is a bialgebra. Moreover, from (1.21) and (1.22) we obtain that p is an algebra map. It remains to show that (C(A), p) is a couniversal solution to the couniversal problem generated by the algebra A and the functor F . Let (B, mB, ηB, ∆B, εB) be a bialgebra and f : B → A be an algebra map. From the universal property of the cofree coalgebra over A there exists an unique coalgebra map g : B → C(A) such that the following diagram commutes: p C(A) / A (1.23) O z< zz g zz zz f zz B We only need to prove that g is also an algebra map. By the argument used above, it is enough to show that: p ◦ g ◦ mB = p ◦ m ◦ (g ⊗ g) and p ◦ g ◦ ηB = p ◦ η (1.24) Having in mind that f is an algebra map, we have:

(1.21) p ◦ m ◦ (g ⊗ g) = m ◦ (p ⊗ p) ◦ (g ⊗ g) = m ◦ (p ◦ g) ⊗ (p ◦ g) (1.23) = m ◦ (f ⊗ f)

= f ◦ mB (1.23) = p ◦ g ◦ mB

(1.23) (1.22) Moreover, p ◦ g ◦ ηB = f ◦ ηB = η = p ◦ η hence (1.24) holds.

Remark 1.3.2 Although in the category of Hopf algebras an epimorphism is not necessarily a surjective map ([53, Corollary 2.2]), k-HopfAlg is a locally presentable category. Indeed, the category k-HopfAlg is cocomplete (Theorem 1.2.9) and has a generator ([120, Corollary 18]). Now, remark that since we work with Hopf algebras over a ﬁeld k, the assumptions from [123, Proposition 4.1.3(d)] are fulﬁlled due to the existence of a left adjoint for the embedding functor F : k-HopfAlg →k-BiAlg. Thus the category k-HopfAlg is locally presentable.

Our main results now follow:

Theorem 1.3.3 The embedding functor F : k − HopfAlg → k − BiAlg has a right adjoint, i.e. the category of Hopf algebras is a coreﬂective subcategory of the category of bialgebras.

26 1.3. REFLECTIVE AND COREFLECTIVE SUBCATEGORIES

Proof: We have already noticed that the category k-HopfAlg is locally presentable. In particular, this implies that k-HopfAlg is colocally small. Moreover, since the category k-HopfAlg is cocomplete (Theorem 1.2.9) and has a generator ([120, Corollary 18]) it follows, by Theo- rem 1.1.7, that the functor F has a right adjoint if and only if it preserves colimits. It can be easily seen from Remark 1.2.4 and Theorem 1.2.9 that F preserves coequalizers and coproducts. Thus F preserves all colimits. Therefore F has a right adjoint, as desired.

Theorem 1.3.4 The forgetful functor F : k − HopfAlg → k − Alg has a right adjoint, i.e. the category k-HopfAlg is a coreﬂective subcategory of k-Alg.

Proof: It follows from Theorem 1.3.1 and Theorem 1.3.3 by composing the right adjoint functors.

Remark 1.3.5 Notice that we can prove directly that the category of Hopf algebras is a coreﬂec- tive subcategory in the category of algebras, by considering the forgetful functor F : k-HopfAlg → k-Alg and using the same arguments as in Theorem 1.3.3. However, we preferred this approach in order to highlight also the existence of the cofree bialgebra on every algebra, result that is just stated by Sweedler in [139].

The construction of the right adjoint for the forgetful functor F : k − HopfAlg → k − Alg was given in [53]. For the sake of completeness we indicate below this construction. Let B be a bialgebra. Consider (P, m, u, ∆, ε), (πn)n∈N the product (in the category BiAlg) B, for n even of the bialgebras (B ) where B = n n∈N n Bop,cop, for n odd By the universal property of the product there is a unique bialgebra map S such that the following diagram commutes for all n ∈ N:

S P op,cop / P

πn+1 πn op,cop Id Bn+1 / Bn

Let H∗(B) be the sum of all subcoalgebras C ⊆ P such that:

c(1)S(c(2)) = uε(c) = S(c(1))c(2)

The pair (H∗(B), β) is a universal solution of the universal problem deﬁned by V and B, where β is the composition of π0 : P → B with the inclusion H∗(B) → P .

27 CHAPTER 1. CATEGORICAL CONSTRUCTIONS

1.4 Monomorphisms of coalgebras and Hopf algebras

In any concrete category C the natural problem of whether epimorphisms are surjective maps arise, as well as the dual problem of whether the monomorphisms are injective maps. This type of problems have already been studied before in several well known categories: for example in [133] it is shown that the property of epimorphisms of being surjective holds in the categories of von Neumann algebras, C∗-algebras, groups, finite groups, Lie algebras, compact groups, while it fails to be true in the categories of finite dimensional Lie algebras, semisimple finite dimensional Lie algebras, locally compact groups and unitary rings (see [82], [137]). The more recent paper [53] deals with the same problems in the context of Hopf algebras: several examples of non-injective monomorphisms and non-surjective epimorphisms are given. It turns out that the above problem is also intimately related to Kaplansky’s first conjecture in the way that every non- surjective epimorphism of Hopf algebras provides a counterexample to Kaplansky’s problem. In [108] the problems mentioned above are studied in the category of coalgebras. The problem of whether epimorphisms of coalgebras are surjective maps is easily dealt with using the existence of a cofree coalgebra on every vector space: hence it is enough to give a positive answer to this problem. The dual problem, on the other hand, is more interesting: an example of a non-injective monomorphism is given and several characterizations of monomorphisms are proved ([108, Theorem 3.5]). In this section we complete the above characterization with two new equivalences. Our interest in this problem comes also from the fact that in the light of [53, Proposition 2.5] which states that a morphism of Hopf algebras is a monomorphism if and only if it is a monomorphism viewed as a morphism of coalgebras it turns out that the same characterization holds for Hopf algebra monomorphisms. Recall from [50] the construction of the trivial coextension of a coalgebra C by a (C,C)- bicomodule N. We define a comultiplication and a counit on the space C o N := C ⊕ N as follows:

∆(c, n) = (c(1), 0) ⊗ (c(2), 0) + (n<−1>, 0) ⊗ (0, n<0>) + (0, n[0]) ⊗ (n[1], 0) ε(c, n) = ε(c) for all (c, n) ∈ C ⊕N. In this way C oN becomes a coalgebra, known as the trivial coextension of C and N. We start by discussing briefly the well known fact that epimorphisms in the category of coalgebras are surjective maps. Recall first that the forgetful functor U : k-CoAlg → kM has a right adjoint, namely the so-called cofree coalgebra on every vector space. Now let f be an epimorphism of coalgebras. Since U is a left adjoint functor it preserves epimorphisms, thus U(f) = f is an epimorphism in kM. Hence f is a surjective map. As mentioned before, a characterization of monomorphisms in k-Coalg is given in [108, The- orem 3.5] and the equivalences (1), (2), (5), (6) in Theorem 1.4.1 are proved there. In what follows we complete the description of monomorphisms in k-Coalg with two more characterizations: the first one, (3), indicates a cohomological description of monomorphisms while the other, (4), is an elementary one involving the cotensor product CDC. However, for the sake of completeness we provide here a complete proof of Theorem 1.4.1.

28 1.4. MONOMORPHISMS OF COALGEBRAS AND HOPF ALGEBRAS

Theorem 1.4.1 Let ϕ : C → D be a coalgebra map. The following statements are equivalent:

1. ϕ is a monomorphism in the category k-CoAlg;

∗ 2. The functor ϕC,D is full;

3. H0(N,C) = H0(N,D) for any (C,C)-bicomodule N;

4. ε(c)d = cε(d) for all c ⊗ d ∈ CDC;

5. The map ηC = ∆C : C → C ⊗ C is surjective;

∗ ∗ 6. The unit of the adjunction (ϕC,D, −DC), η : 1MC → (−DC) ◦ ϕC,D is a natural isomorphism.

Proof: The equivalence 2. ⇔ 6. is obvious, as well as the implication 6. ⇒ 5. 1. ⇒ 3. Suppose ϕ is a monomorphism of coalgebras. It is easy to see that H0(N,C) ⊂ H0(N,D). 0 Now let γ ∈ H (N,D). Deﬁne the maps πC , β : C o N → C by:

πC (c, n) = c

β(c, n) = c − n<−1>γ(n<0>) + γ(n[0])n[1] for all (c, n) ∈ C o N. We will prove that both πC and β are coalgebra maps. For every (c, n) ∈ C o N we have: εC ◦ πC (c, n) = εC (c) = ε(c, n) and

(πC ⊗ πC )(∆(c, n)) = (πC ⊗ πC ) (c(1), 0) ⊗ (c(2), 0) + (n<−1>, 0) ⊗ (0, n<0>) + +(0, n[0]) ⊗ (n[1], 0)

= c(1) ⊗ c(2) + n<−1> ⊗ 0 + 0 ⊗ n[0]

= c(1) ⊗ c(2)

= ∆C ◦ πC (c, n)

Thus πC is a coalgebra map. Moreover, for β we obtain:

εC ◦ β(c, n) = εC (c − n<−1>γ(n<0>) + γ(n[0])n[1])

= εC (c) − εC (n<−1>)γ(n<0>) + γ(n[0])εC (n[1]) = εC (c) − γ εC (n<−1>)n<0> +γ n[0]εC (n[1]

= εC (c) − γ(n) + γ(n) = ε(c, n)

29 CHAPTER 1. CATEGORICAL CONSTRUCTIONS and (β ⊗ β) ◦ ∆(c, n) = (β ⊗ β) (c(1), 0) ⊗ (c(2), 0) + (m<−1>, 0) ⊗ (0, m<0>) + +(0, m[0]) ⊗ (m[1], 0) = β (c(1), 0) ⊗β (c(2), 0) +β (m<−1>, 0) ⊗β (0, m<0>) + +β (0, m[0]) ⊗β (m[1], 0) = c(1) ⊗ c(2) + n<−1> ⊗ −n<0><−1>γ(n<0><0>) + +γ(n<0>[0])n<0>[1] + −n[0]<−1>γ(n[0]<0>) + γ(n[0][0])n[0][1] ⊗n[1]

= c(1) ⊗ c(2) − n<−1> ⊗ n<0><−1>γ(n<0><0>) + γ(n[0][0])n[0][1] ⊗ n[1]

= c(1) ⊗ c(2) − n<−1>(1) ⊗ n<0>(2)γ(n<0>) + γ(n[0])n[1](1) ⊗ n[1](2) = c(1) ⊗ c(2) − ∆C n<−1>γ(n<0>) +∆C γ(n[0])n[1] = ∆C c − n<−1>γ(n<0>) + γ(n[0])n[1]

= ∆C ◦ β(c, n)

Hence, we proved that πC and β are coalgebra maps. Furthermore, it is easy to see that ϕ◦πC = ϕ ◦ β and since we assumed that ϕ is a monomorphism it follows from here that πC = β. Thus 0 n<−1>γ(n<0>) = γ(n[0])n[1] which implies that γ ∈ H (N,C). 2. ⇒ 1. Let α, β : E → C coalgebra maps such that ϕ ◦ α = ϕ ◦ β := g. We denote the comultiplications on C and E respectively, as follows: ∆C (c) = c(1) ⊗ c(2) and ∆E = c{1} ⊗ c{2}. As we noticed in the previous section, C can be endowed with a right D-comodule structure ρ : C → C ⊗ D via the coalgebra map ϕ: ρ(c) = c(1) ⊗ ϕ(c(2)). Similarly, we endow the coalgebra E with a right D-comodule structure ψ : E → E ⊗ D via the coalgebra map g: ψ(c) = c{1} ⊗ g(c{2}). Since α is a coalgebra map we obtain: α(x)(1) ⊗ ϕ α(x)(2) = α(x{1}) ⊗ ϕ α(x{2})

Having in mind that ϕ ◦ α = ϕ◦, the above equation can be rewritten as: α(x)(1) ⊗ ϕ α(x)(2) = α(x{1}) ⊗ ϕ β(x{2})

∗ hence α is a morphism of right D-comodules. Since ϕC,D is a full functor it follows that α is a morphism of right C-comodules. Thus, we have:

α(x{1}) ⊗ α(x{2}) = α(x{1}) ⊗ β(x{2}) for all x ∈ E. By applying εC ⊗ I to the above identity we obtain α(x) = β(x) for all x ∈ E, as desired. l 3. ⇒ 4.C DC is a (C,C)-bicomodule with left and right structures given by: ψ (a⊗b) = CDC a ⊗ a ⊗ b and ψr (a ⊗ b) = a ⊗ b ⊗b for all a ⊗ b ∈ C C. Deﬁne (1) (2) C DC (1) (2) D ∗ T : CDC → k, T (m⊗n) = ε(m)ε(n). Remark that T ∈ CDC . Now let a⊗b ∈ CDC, that is a(1) ⊗ ϕ(a(2)) ⊗ b = a ⊗ ϕ(b(1)) ⊗ b(2). By applying ε ⊗ I ⊗ ε in the above identity we 0 0 obtain ϕ(a)ε(b) = ε(a)ϕ(b). Thus T ∈ H (CDC,D) = H (CDC,C) and it follows that

30 1.4. MONOMORPHISMS OF COALGEBRAS AND HOPF ALGEBRAS aε(b) = ε(a)b. 5. ⇒ 4. Let a ⊗ b ∈ CDC. Since ∆C = ηC is surjective, there exists an element c ∈ C such that ∆C (c) = a ⊗ b. Hence c(1) ⊗ c(2) = a ⊗ b and we obtain: ε(a)b = c = aε(b). C 4. ⇒ 2. Let M ∈ M . Deﬁne νM : MDC → M by νM (m ⊗ c) = mε(c). For any m ⊗ c ∈ MDC we have: (ν ◦ I) ◦ ρr (m ⊗ c) = (ν ◦ I)(m ⊗ c ⊗ c ) M MDC M (1) (2) = νM (m ⊗ c(1)) ⊗ c(2)

= mε(c(1)) ⊗ c(2)

= m[0]ε(m[1]) ⊗ c

= m[0] ⊗ ε(m[1])c

= m[0] ⊗ m[1]ε(c) r = ρM (mε(c)) r = (ρM ◦ νM )(m ⊗ c) where we used the fact that m[1] ⊗ c ∈ CDC for all m ⊗ c ∈ MDC. Thus νM is a morphism of right C-comodules. Moreover, in the computations above we also proved that r C (ρM ◦ νM )(m ⊗ c) = m ⊗ c for all m ⊗ c ∈ MDC. Hence, we proved that for all M ∈ M r there exists a morphism of right C-comodules such that ρM ◦ νM = I. Having in mind that r ∗ ηM = ρM , as remarked before, this is enough to prove that ϕC,D is a full functor.

In view of a remark from [53], a morphism of Hopf algebras is a monomorphism if and only if it is a monomorphism viewed as a morphism of coalgebras. Thus, we obtain the following useful fact:

Corollary 1.4.2 Let ϕ : K → L be a Hopf algebra map. The following are equivalent:

1) ϕ : K → L is a Hopf algebra monomorphism;

2) The map ηK = ∆K : K → KLK is surjective (hence is bijective); 3) H0(N,K) = H0(N,L) for any (K,K)-bicomodule N;

P i i P i i P i i 4) i∈I ε(x )y = i∈I x ε(y ) for all i∈I x ⊗ y ∈ KLK.

Example 1.4.3 Let π : M2(k) → M2(k)/I be the canonical projection, where k is a ﬁeld and 2 I is the coideal of the comatrix coalgebra M (k) generated by the elements c21. It is proved in [108], using a result on epimorphisms of ﬁnite dimensional algebras [108, Theorem 3.2], that π is a non-injective monomorphism of coalgebras. However, this can be easily shown by a simple computation using 7) of Theorem 1.4.1.

31 CHAPTER 1. CATEGORICAL CONSTRUCTIONS

1.5 Braidings on the category of bimodules

Braided monoidal categories play a key role in several areas of mathematics like quantum groups, noncommutative geometry, knot theory, quantum field theory and 3-manifolds. It is well-known that the category AMA of bimodules over an algebra A over a commutative ring k is monoidal. The aim of this section is to give an answer to the following natural question: given an algebra A, describe all braidings on AMA. Besides the purely categorical significance this problem is also relevant in noncommutative geometry where braidings on AMA are used to develop the theory of wedge products of differential forms or connections on bimodules. The question is not as obvious as it seems: a first attempt might be to use the switch map to define the braiding, but this is not well-defined, even in the case when A is a commutative algebra. However, there are non-trivial examples of braidings on the category of bimodules. For example, let A = Mn(k) be a matrix algebra; then cM,N : M ⊗A N → N ⊗A M given by the formula

n X cM,N (m ⊗A n) = eij n eti ⊗A m ejt i,j,t=1 is a braiding on the category of A-bimodules (see Example 1.5.9 for full detail). A ﬁrst general result is Theorem 1.5.1, stating that braidings on the category of A-bimodules are in bijective correspondence with canonical R-matrices, these are invertible elements R in the threefold tensor product A ⊗ A ⊗ A, satisfying a list of axioms. In this situation, we will say that (A, R) is an algebra with a canonical R-matrix. Actually, this result is inspired by a classical result of Hopf algebras: braidings on the category of (left) modules over a bialgebra H are in one-to-one correspondence with quasitriangular structures on H, these are elements R in the two-fold tensor product H ⊗ H satisfying certain properties. We refer to [106, Theorem 10.4.2] for detail. The next step is to reduce the list of axioms to two equations, a centralizing condition and a normalizing condition, and then we can prove in Theorem 1.5.2 that all braidings on a category of bimodules are symmetries. In the situation where A is commutative, we have a complete classiﬁcation: A admits a canonical R-matrix R if and only if k → A is an epimorphism in the category of rings, and then R is trivial, see Proposition 1.5.3. A The invariants functor G = (−) : AMA → Mk has a left adjoint F = A ⊗ −. We prove that G is a separable functor [109, 131] if and only if G is fully faithful and this implies that A admits a canonical R-matrix. The converse property also holds if A is free as a k-module, and then the braiding on the category of A-bimodules is unique, cf. Theorem 1.5.6. Azumaya algebras were introduced in [28] under the name central separable algebras; a more restrictive class was considered earlier by Azumaya in [29]. Azumaya algebras are the proper generalization of central simple algebras to commutative rings. The Brauer group consists of the set of Morita equivalence classes of Azumaya algebras. There exists a large literature on Azumaya algebras and the Brauer group, see for example the reference list in [45]. A is an Azu- maya algebra if and only if G is an equivalence of categories, and then G is separable. Therefore the category of bimodules over an Azumaya algebra is braided monoidal, that is any Azumaya algebra admits a canonical R-matrix. R can be described explicitly in the cases where A is a matrix algebra or a quaternion algebra, see Examples 1.5.9 and 1.5.10. Not every algebra with a canonical R-matrix is Azumaya; for example Q is not a Z-Azumaya algebra, but 1 ⊗ 1 ⊗ 1 is

32 1.5. BRAIDINGS ON THE CATEGORY OF BIMODULES a canonical R-matrix, since Z → Q is an epimorphism of rings. Thus algebras with a canonical R-matrix can be viewed as generalizations of Azumaya algebras. Applying Theorem 1.5.6 to finite dimensional algebras over fields, we obtain a new characterization of central simple algebras, namely central simple algebras are the finite dimensional algebras admitting a canonical R-matrix. As a final application, we construct a simultaneous solution of the quantum Yang-Baxter equation and the braid equation from any canonical R-matrix, see Theorem 1.5.13.

Let A be an algebra over a commutative ring k and AMA = (AMA, − ⊗A −,A) the monoidal category of A-bimodules. A(n) will be a shorter notation for the n-fold tensor product A ⊗ · · · ⊗ (3) 1 2 3 A, where ⊗ = ⊗k. An element R ∈ A will be denoted by R = R ⊗ R ⊗ R , where summation is implicitly understood. Our ﬁrst aim is to investigate braidings on AMA.

Theorem 1.5.1 Let A be a k-algebra. Then there is a bijective correspondence between the 1 2 3 (3) class of all braidings c on AMA and the set of all invertible elements R = R ⊗R ⊗R ∈ A satisfying the following conditions, for all a ∈ A:

R1 ⊗ R2 ⊗ aR3 = R1a ⊗ R2 ⊗ R3 (1.25) aR1 ⊗ R2 ⊗ R3 = R1 ⊗ R2a ⊗ R3 (1.26) R1 ⊗ aR2 ⊗ R3 = R1 ⊗ R2 ⊗ R3a (1.27) R1 ⊗ R2 ⊗ 1 ⊗ R3 = r1R1 ⊗ r2 ⊗ r3R2 ⊗ R3 (1.28) R1 ⊗ 1 ⊗ R2 ⊗ R3 = R1 ⊗ R2r1 ⊗ r2 ⊗ R3r3 (1.29) where r = r1 ⊗ r2 ⊗ r3 = R. Under the above correspondence the braiding c corresponding to R is given by the formula

1 2 3 cM,N : M ⊗A N → N ⊗A M, cM,N (m ⊗A n) = R nR ⊗A mR . (1.30) for all M, M ∈ AMA, m ∈ M and n ∈ N. An invertible element R ∈ A(3) satisfying (1.25)-(1.29) is called a canonical R-matrix and (A, R) is called an algebra with a canonical R-matrix.

Proof: A(2) is an A-bimodule via the usual actions a(x ⊗ y)b = ax ⊗ yb, for all a, b, x, y ∈ A. Let c : AMA × AMA → AMA × AMA be a braiding on AMA. For each M,N ∈ AMA, we have an A-bimodule isomorphism cM,N : M ⊗A N → N ⊗A M, that is natural in M and N. Now consider

(3) ∼ (2) (2) (3) ∼ (2) (2) cA(2),A(2) : A = A ⊗A A → A = A ⊗A A , and let R = cA(2),A(2) (1⊗1⊗1). c is completely determined by R. For M,N ∈ AMA, m ∈ M (2) (2) and n ∈ N, we consider the A-bimodule maps fm : A → M and gn : A → N given by the formulas fm(a ⊗ b) = amb and gn(a ⊗ b) = anb. From the naturality of c, it follows that

(gn ⊗A fm) ◦ cA(2),A(2) = cM,N ◦ (fm ⊗A gn).

33 CHAPTER 1. CATEGORICAL CONSTRUCTIONS

(1.30) follows after we evaluate this formula at 1 ⊗ 1 ⊗ 1. Obviously cM,N (ma ⊗A n) = cM,N (m ⊗A an). Furthermore, cM,N (am ⊗A n) = acM,N (a ⊗A n) and cM,N (m ⊗A na) = cM,N (a ⊗A n)a since cM,N is a bimodule map. If we write these three formulas down in the case where M = N = A(2), and m = n = 1 ⊗ 1, then we obtain (1.25)-(1.27). c satisﬁes the two triangle equalities

cM⊗AN,P = (cM,P ⊗A N) ◦ (M ⊗A cN,P );

cM,N⊗AP = (N ⊗A cM,P ) ◦ (cM,N ⊗A P ).

The ﬁrst equality is equivalent to

1 2 3 1 1 2 2 3 3 R pR ⊗A m ⊗A nR = r R pR r ⊗A mr ⊗A nR , for all m ∈ M, n ∈ N and p ∈ P . If we take M = N = P = A(2) and m = n = p = 1 ⊗ 1, then we ﬁnd that R1 ⊗ R2 ⊗ 1 ⊗ R3 = r1R1 ⊗ R2r2 ⊗ r3 ⊗ R3. Applying (1.27), we ﬁnd that (1.28) holds. In a similar way, the second triangle equality implies (1.29). We can apply the same arguments to the inverse braiding c−1. This gives S = S1 ⊗ S2 ⊗ S3 = c−1 (1 ⊗ 1 ⊗ 1). Then we have that A(2),A(2)

−1 −1 1 2 3 1 3 2 1 2 3 m ⊗A n = (cN,M ◦ cM,N )(m ⊗A n) = cN,M (R nR ⊗A mR ) = S mR S ⊗A R nR S .

Now take m = n = 1 ⊗ 1 ∈ A(2). Then we ﬁnd

(1.26) 1 ⊗ 1 ⊗ 1 = S1 ⊗ R3S2R1 ⊗ R2S3 = R1S1 ⊗ R3S2 ⊗ R2S3 (1.27) (1.25) (1.26) = R1S1 ⊗ S2 ⊗ R2S3R3 = R1S1R2 ⊗ S2 ⊗ S3R3 = S1R2 ⊗ S2R1 ⊗ S3R3.

In a similar way, we have that R1S2 ⊗R2S1 ⊗R3S3 = 1⊗1⊗1, and it follows that S2 ⊗S1 ⊗S3 is the inverse of R1 ⊗ R2 ⊗ R3. Conversely, assume that R ∈ A(3) is invertible and satisﬁes (1.25)-(1.29). Then we deﬁne c using (1.30). Straightforward computations show that c is a braiding on AMA.

Let c be a braiding on AMA and R the corresponding canonical R-matrix. Then c is a symmetry, if and only if S = R, this means that

R2r1 ⊗ R1r2 ⊗ R3r3 = 1 ⊗ 1 ⊗ 1 (1.31) that is, R−1 = R2 ⊗ R1 ⊗ R3. The next theorem shows that the list of equations satisﬁed by an R-matrix from Theorem 1.5.1 can be reduced to two equations and furthermore, we prove that all braidings on the category of A-bimodules are symmetries.

Theorem 1.5.2 Let A be a k-algebra. Then there is a bijection between the set of canonical R-matrices and the set of all elements R ∈ A(3) satisfying (1.27) and the normalizing condition

R1R2 ⊗ R3 = R2 ⊗ R3R1 = 1 ⊗ 1 (1.32)

34 1.5. BRAIDINGS ON THE CATEGORY OF BIMODULES

Furthermore, in this situation, R is invariant under cyclic permutation of the tensor factors,

R = R2 ⊗ R3 ⊗ R1 = R3 ⊗ R1 ⊗ R2, (1.33) and we have the additional normalizing condition

R1 ⊗ R2R3 = 1 ⊗ 1. (1.34)

In particular, every braiding on AMA is a symmetry.

Proof: Let R be an R-matrix as in Theorem 1.5.1, i.e. R is invertible and satisfies (1.25)-(1.29). Multiplying the second and the third tensor factor in (1.29), we find that R = R1 ⊗ R2r1r2 ⊗ R3r3 = R(1 ⊗ r1r2 ⊗ r3). From the fact that R is invertible, it follows that 1 ⊗ 1 ⊗ 1 = 1 ⊗ r1r2 ⊗ r3, and the first normalizing condition of (1.32) follows after we multiply the first two tensor factors. On the other hand, if we apply the flip map on the last two positions in (1.28) we obtain that R1 ⊗ R2 ⊗ R3 ⊗ 1 = r1R1 ⊗ r2 ⊗ R3 ⊗ r3R2. Multiplying the last two positions we obtain: (1.25) R = r1R1 ⊗ r2 ⊗ R3r3R2 = r1R3R1 ⊗ r2 ⊗ r3R2 = R(R3R1 ⊗ 1 ⊗ R2)

As R is invertible it follows that R3R1 ⊗ R2 = 1 ⊗ 1, as needed. Conversely, assume now that R satisﬁes (1.27) and (1.32). We will show that R is a canonical R-matrix satisfying (1.31) and hence from the observations preceding Theorem 1.5.2 we obtain that the braiding c corresponding to R is a symmetry. First we show that R is invariant under cyclic permutation of the tensor factors. (1.32) (1.27) R3 ⊗ R1 ⊗ R2 = R3r1r2 ⊗ r3R1 ⊗ R2 = R3r2 ⊗ r3R1 ⊗ r1R2 (1.27) (1.32) = r2 ⊗ r3R3R1 ⊗ r1R2 = r2 ⊗ r3 ⊗ r1.

This implies immediately that the central conditions (1.25)-(1.26) are also satisﬁed. Next we show that (1.28)-(1.29) are satisﬁed. (1.26) r1R1 ⊗ r2 ⊗ r3R2 ⊗ R3 = R1 ⊗ r2 ⊗ r3R2r1 ⊗ R3 (1.27) (1.32) = R1 ⊗ R2r2 ⊗ r3r1 ⊗ R3 = R1 ⊗ R2 ⊗ 1 ⊗ R3; (1.27) R1 ⊗ R2r1 ⊗ r2 ⊗ R3r3 = R1 ⊗ r3R2r1 ⊗ r2 ⊗ R3 (1.27) (1.32) = R1 ⊗ r3r1 ⊗ R2r2 ⊗ R3 = R1 ⊗ 1 ⊗ R2 ⊗ R3.

Finally, we are left to prove that R is invertible and (1.31) holds. Indeed, we have: (1.26) R(r2 ⊗ r1 ⊗ r3)=R1r2 ⊗ R2r1 ⊗ R3r3 = R1r2R2 ⊗ r1 ⊗ R3r3 (1.27) (1.32) (1.33) (1.32) = R1R2 ⊗ r1 ⊗ R3r2r3 = 1 ⊗ r1 ⊗ r2r3 = 1 ⊗ r3 ⊗ r1r2 = 1 ⊗ 1 ⊗ 1.

In the same way it can be proved that (r2 ⊗r1 ⊗r3)R = 1⊗1⊗1. Moreover, the last computation also shows that (1.31) holds and the proof is now ﬁnished.

The commutative case is completely classiﬁed by the following result.

35 CHAPTER 1. CATEGORICAL CONSTRUCTIONS

Proposition 1.5.3 Let A be a k-algebra. Then:

1. If a monomial x ⊗ y ⊗ z is a canonical R-matrix, then it is equal to 1 ⊗ 1 ⊗ 1.

2. 1 ⊗ 1 ⊗ 1 is a canonical R-matrix if and only if uA : k → A is an epimorphism of rings. 3. If A is commutative, then (A, R) is an algebra with a canonical R-matrix if and only if R = 1 ⊗ 1 ⊗ 1 and uA : k → A is an epimorphism in the category of rings.

Proof: 1. Let R = x ⊗ y ⊗ z be a canonical R-matrix. From (1.28)-(1.29), it follows that x ⊗ 1 ⊗ y ⊗ z = x ⊗ yx ⊗ y ⊗ z2 and x ⊗ y ⊗ 1 ⊗ z = x2 ⊗ y ⊗ zy ⊗ z. Since R is invertible, this implies that 1 ⊗ 1 ⊗ 1 ⊗ 1 = 1 ⊗ yx ⊗ 1 ⊗ z and 1 ⊗ 1 ⊗ 1 ⊗ 1 = x ⊗ 1 ⊗ zy ⊗ 1, and, multiplying tensor factors, we ﬁnd that 1⊗1 = yx⊗z and 1⊗1 = x⊗zy. It then follows that yxz = xzy = 1, hence y is invertible with y−1 = xz. Finally (1.27) x ⊗ y ⊗ z = x ⊗ y2y−1 ⊗ z = x ⊗ yy−1 ⊗ zy = 1 ⊗ 1 ⊗ 1. 2. If R = 1 ⊗ 1 ⊗ 1, then the three centralizing conditions (1.25)-(1.27) are equivalent to a ⊗ 1 = 1 ⊗ a, for all a ∈ A, which is equivalent to uA : k → A being an epimorphism of rings, see [137]. 3. Assume that (A, R) is an algebra with a canonical R-matrix. Then: (1.28) R1 ⊗ R2 ⊗ 1 ⊗ R3 = r1R1 ⊗ r2 ⊗ r3R2 ⊗ R3 (1.27) X = r1R1 ⊗ R2r2 ⊗ r3 ⊗ R3 = R1r1 ⊗ R2r2 ⊗ r3 ⊗ R3. At the third step, we used the fact that A is commutative. From the fact that R is invertible, it follows that R1 ⊗ R2 ⊗ 1 ⊗ R3 = 1 ⊗ 1 ⊗ 1 ⊗ 1 and R = 1 ⊗ 1 ⊗ 1. The rest of the proof follows from 2.

Remarks 1.5.4 1. The notion of quasi-triangular bialgebroid was introduced in [64, Def. 19]. Quasi-triangular structures on a bialgebroid are given by universal R-matrices, see [64, Prop. 20] and [37, Def. 3.15], and correspond bijectively to braidings on the category of modules over the bialgebroid [37, Theorem 3.16]. It is well-known that Ae is an A-bialgebroid, with the Sweedler canonical coring as underlying coring, and A-bimodules are left Ae-modules. Comparing our Theorem 1.5.1 with the (left handed) version of [37, Theorem 3.16] yields that canonical R-matrices for A correspond bijectively to universal R-matrices for the canonical bialgebroid Ae. This leads to an alternative proof of Theorem 1.5.1, if we identify the R-matrices from [64] with our R-matrices. This, however, is more complicated than the straightforward proof that we presented, that also has the advantage that it is self-contained and avoids all technicalities on bialgebroids. 2. (1.28)-(1.29) can be rewritten as R124 = R123R134 and R134 = R124R234 in the algebra A(4). 3. It follows from Proposition 1.5.3 that there is only one braiding on the category of (left) k-modules, namely the one given by the usual switch map.

36 1.5. BRAIDINGS ON THE CATEGORY OF BIMODULES

Before we state our next main result Theorem 1.5.6, we need a technical Lemma. If M ∈ AMA, then A ⊗ M is a k ⊗ A-bimodule, and we can consider

k⊗A X X X (A ⊗ M) = { ai ⊗ mi ∈ A ⊗ M | ai ⊗ ami = ai ⊗ mia, for all a ∈ A}. i i i

If M = A(2), then (A ⊗ A(2))k⊗A is the set of elements R ∈ A(3) satisfying (1.27). We have a A k⊗A map αM : A ⊗ M → (A ⊗ M) , αM (a ⊗ m) = a ⊗ m.

Lemma 1.5.5 Let M be an A-bimodule. The map αM is injective if A is ﬂat as a k-module, and bijective if A is free as a k-module.

A Proof: If A is ﬂat, then A ⊗ M → A ⊗ M is injective, and then αM is also injective. Assume that A is free as a k-module, and let {ej | j ∈ I} be a free basis of A. Assume that P k⊗A P j j x = i ai ⊗ mi ∈ (A ⊗ M) . For all i, we can write ai = j∈I αi ej, for some αi ∈ k. P P j Then x = j∈I ej ⊗ i αi mi . Now

X X j X X X X j x = ej ⊗ αi ami = ai ⊗ ami = ai ⊗ mia = ej ⊗ αi mia , j∈I i i i j∈I i

P j P j P j A hence i αi ami = i αi mia, for all j ∈ I, and i αi mi ∈ M . We conclude that x = P P j j∈I ej ⊗ i αi mi ∈ Im αM , and this shows that αM is surjective.

In our next result we assume A to be ﬂat over k. Hence the map αA(2) is injective and this will allows us to identify the elements in A ⊗ (A ⊗ A)A with the elements in A ⊗ A ⊗ A satisfying (1.27). We also recall from the proof of Theorem 1.5.2 that the elements R ∈ A ⊗ A ⊗ A satisfying (1.27) and (1.32) are invertible with the inverse given by R−1 = R2 ⊗ R1 ⊗ R3.

Theorem 1.5.6 Let A be a ﬂat k-algebra and consider the conditions:

1. (F,G) is a pair of inverse equivalences, that is, A is an Azumaya algebra;

A 2. The functor G = (−) : AMA → Mk is fully faithful;

A 3. the functor G = (−) : AMA → Mk is separable; 4. there exists R = R1 ⊗ R2 ⊗ R3 ∈ A ⊗ (A ⊗ A)A such that R1R2 ⊗ R3 = 1 ⊗ 1;

5. there exists a unique R = R1 ⊗ R2 ⊗ R3 ∈ A ⊗ (A ⊗ A)A such that R1R2 ⊗ R3 = 1 ⊗ 1;

(3) 6. there exists a braiding on AMA, that is, there exists R ∈ A such that (A, R) is an algebra with a canonical R-matrix.

Then (1) ⇒ (2) ⇔ (3) ⇔ (4) ⇔ (5) ⇒ (6). If A is central, then (2) ⇒ (1). If A is free as a k-module, then (6) ⇒ (5), and in this case the braiding on AMA is unique. If k is a ﬁeld, and A is ﬁnite dimensional, then (6) ⇒ (1), and all six assertions are equivalent.

37 CHAPTER 1. CATEGORICAL CONSTRUCTIONS

Proof: (1) ⇒ (2), (2) ⇒ (3) and (5) ⇒ (4) are trivial. (3) ⇒ (4). If G is separable, then we have a natural transformation ζ : 1 ⇒ FG such that 1 2 3 (2) εM ◦ ζM = M, for all M ∈ AMA. Now let R = ζA(2) (1 ⊗ 1) = R ⊗ R ⊗ R ∈ FG(A ) = A 1 2 3 A ⊗ (A ⊗ A) . Then 1 ⊗ 1 = (εA(2) ◦ ζA(2) )(1 ⊗ 1) = R R ⊗ R . The natural transformation ζ is completely determined by R. For an A-bimodule M and m ∈ M, we deﬁne fm as in the proof of Theorem 1.5.1. From the naturality of ζ, it follows that the diagram ζ A(2) A(2) / A ⊗ (A ⊗ A)A

A fm A⊗(fm)

ζM M / A ⊗ M A commutes. Evaluating the diagram at 1 ⊗ 1, we ﬁnd that

1 2 3 ζM (m) = R ⊗ R mR . (1.35)

P A 2 3 1 (4) ⇒ (6). Write R = i ai ⊗ bi, with ai ∈ A and bi ∈ (A ⊗ A) . Then R ⊗ R R = P P 1 2 3 i biai = i aibi = R R ⊗ R = 1 ⊗ 1, thus R satisfies (1.32). Moreover, as R ∈ A ⊗ (A ⊗ A)A it follows that R also satisfies (1.27). Then, using Theorem 1.5.2 we obtain that R is a canonical R-matrix and it determines a braiding on AMA. (4) ⇒ (2). Given R ∈ A ⊗ (A ⊗ A)A satisfying R1R2 ⊗ R3 = 1 ⊗ 1, we define ζ using (1.35). 1 2 3 1 2 3 It follows immediately that (εM ◦ ζM )(m) = ε(R ⊗ R mR ) = R R mR = m. Moreover, it follows as in the proof of 4) ⇒ 6) that (1.26) and (1.34) are satisfied. For ai ∈ A A and mi ∈ M , we then compute

X X 1 2 3 (1.26) X 1 2 3 (ζM ◦ εM )( ai ⊗ mi) = R ⊗ R aimiR = aiR ⊗ R miR i i i X 1 2 3 (1.34) X = aiR ⊗ R R mi = ai ⊗ mi. i i This shows that ε is a natural transformation with inverse ζ, and G is fully faithful. (2) ⇒ (5). We have already seen that 2) implies 4), and this shows that R exists. If G is A fully faithful, then εM is invertible, for all M ∈ AMA. If R ∈ A ⊗ (A ⊗ A) satisfies 1 2 3 −1 R R ⊗ R = 1 ⊗ 1, then εA⊗A(R) = 1 ⊗ 1, hence R = εA⊗A(1 ⊗ 1), i.e. R is unique. (6) ⇒ (4). From (5), it follows that there exists R ∈ (A⊗A(2))k⊗A such that R1R2⊗R3 = 1⊗1, −1 A see Theorem 1.5.2. α (2) is bijective, see Lemma 1.5.5, hence α (R) ∈ A⊗(A⊗A) satisfies A A(2) (3). The uniqueness of R follows from (4). A (4) ⇒ (1). Assume that A is central. From (4), it follows that εA⊗A : A ⊗ (A ⊗ A) → A ⊗ A is surjective, and then it follows from [28, Theorem 3.1] that A is separable over Z(A) = k. Thus A is central separable, and therefore Azumaya. (6) ⇒ (1). If k is a field, then A is free, so (6) implies (5), and, a fortiori, (2). Then εA : A ⊗ AA → A is an isomorphism of A-bimodules, and therefore also of vector spaces. A count A of dimensions shows that dimk(Z(A)) = dimk(A ) = 1, so that Z(A) = k1A, and A is central, and then (1) follows from (2).

38 1.5. BRAIDINGS ON THE CATEGORY OF BIMODULES

In particular, applying Theorem 1.5.6 for ﬁnite dimensional algebras over ﬁelds we obtain the following surprising characterization of central simple algebras:

Corollary 1.5.7 For a ﬁnite dimensional algebra A over a ﬁeld k, the following assertions are equivalent:

1. A is a central simple algebra;

2. there exists a (unique) braiding on AMA; 3. there exists a (unique) invertible element R ∈ A ⊗ A ⊗ A satisfying the conditions R1 ⊗ a R2 ⊗ R3 = R1 ⊗ R2 ⊗ R3 a and R1R2 ⊗ R3 = R2 ⊗ R3R1 = 1 ⊗ 1, for all a ∈ A.

For any k-algebra A, the functor F : Mk → AMA is strong monoidal. Indeed, for any 0 N,N ∈ Mk, we have natural isomorphisms ϕ0 : F (k) = A ⊗ k → A and

0 0 0 0 ϕN,N 0 : F (N) ⊗A F (N ) = (A ⊗ N) ⊗A (A ⊗ N ) → F (N ⊗ N ) = A ⊗ N ⊗ N satisfying all the necessary axioms, see [81].

Proposition 1.5.8 Let (A, R) be an algebra with a canonical R-matrix. Then the symmetry on AMA is such that the functor F : Mk → AMA preserves the symmetry.

Proof: We have to show that the following diagram commutes

ϕ 0 0 N,N 0 (A ⊗ N) ⊗A (A ⊗ N ) / A ⊗ N ⊗ N

cA⊗N,A⊗N0 A⊗τN,N0 ϕ 0 0 N ,N 0 (A ⊗ N ) ⊗A (A ⊗ N) / A ⊗ N ⊗ N

0 0 0 Here τN,N 0 : N ⊗ N → N ⊗ N is the usual switch map. For a, b ∈ A, n ∈ N and n ∈ N, we compute

0 (1.30) 1 2 0 3 (ϕN 0,N ◦ cA⊗N,A⊗N 0 )((a ⊗ n) ⊗A (b ⊗ n )) = ϕN 0,N (R bR ⊗ n ) ⊗A aR ⊗ n (1.25) = R1bR2aR3 ⊗ n0 ⊗ n = R1R2abR3 ⊗ n0 ⊗ n (1.32) 0 0 = ab ⊗ n ⊗ n = ab ⊗ τN,N 0 (n ⊗ n ) 0 = ((A ⊗ τN,N 0 ) ◦ ϕN,N 0 )((a ⊗ n) ⊗A (b ⊗ n )) and the proof is ﬁnished.

If A is an Azumaya algebra, then it follows from Theorem 1.5.6 that we have a symmetry on the category of A-bimodules AMA. In Example 1.5.9 and Example 1.5.10, we give explicit formulas for R in the case where A is a matrix ring or a quaternion algebra; in both cases A is free, so that the R-matrix is unique.

39 CHAPTER 1. CATEGORICAL CONSTRUCTIONS

Example 1.5.9 Let A = Mn(k) be a matrix algebra. Then

n X R = eij ⊗ eki ⊗ ejk, i,j,k=1 where eij is the elementary matrix with 1 in the (i, j)-position and 0 elsewhere. Indeed, For all indices i, j, p, q, we have

n n X X epq( eki ⊗ ejk) = epi ⊗ ejq = ( eki ⊗ ejk)epq, k=1 k=1

Pn A Pn Pn A hence k=1 eki ⊗ ejk ∈ (A ⊗ A) and R = i,j=1 eij ⊗ ( k=1 eki ⊗ ejk) ∈ A ⊗ (A ⊗ A) . Finally n X X eijeki ⊗ ejk = eii ⊗ ejj = 1 ⊗ 1. i,j,k=1 i,j=1n

Example 1.5.10 Let K be a commutative ring, such that 2 is invertible in K, and take two invertible elements a, b ∈ K. The generalized quaternion algebra A = aKb is the free K- module with basis {1, i, j, k} and multiplication deﬁned by

i2 = a, j2 = b, ij = −ji = k.

It is well-known that A is an Azumaya algebra. The corresponding R-matrix is 1 1 R = (1 ⊗ 1 ⊗ 1) + (1 ⊗ i ⊗ i + i ⊗ 1 ⊗ i + i ⊗ i ⊗ 1) 4 4a 1 1 + (1 ⊗ j ⊗ j + j ⊗ 1 ⊗ j + j ⊗ j ⊗ 1) − (1 ⊗ k ⊗ k + k ⊗ 1 ⊗ k + k ⊗ k ⊗ 1) 4b 4ab 1 1 + (i ⊗ j ⊗ k + j ⊗ k ⊗ i + k ⊗ i ⊗ j) − (j ⊗ i ⊗ k + k ⊗ j ⊗ i + i ⊗ k ⊗ j) 4ab 4ab It is easy to show that R satisﬁes (1.27) and (1.32). Indeed, 1 1 1 1 1 1 1 R1 R2 ⊗ R3 = 1 ⊗ 1 + i ⊗ i + j ⊗ j − k ⊗ k + i ⊗ i + 1 ⊗ 1 + k ⊗ k − 4 4a 4b 4ab 4a 4 4ab 1 1 1 1 1 1 1 j ⊗ j + j ⊗ j + 1 ⊗ 1 + k ⊗ k − i ⊗ i − k ⊗ k + 1 ⊗ 1 − 4b 4b 4 4ab 4a 4ab 4 1 1 j ⊗ j − i ⊗ i = 1 ⊗ 1 4b 4a

1 1 1 1 1 1 1 R2 ⊗ R3 R1 = 1 ⊗ 1 + i ⊗ i + j ⊗ j − k ⊗ k + 1 ⊗ 1 + i ⊗ i − j ⊗ j + 4 4a 4b 4ab 4 4a 4b 1 1 1 1 1 1 1 k ⊗ k + 1 ⊗ 1 + j ⊗ j − i ⊗ i + k ⊗ k + 1 ⊗ 1 − k ⊗ k − 4ab 4 4b 4a 4ab 4 4ab 1 1 i ⊗ i − j ⊗ j = 1 ⊗ 1 4a 4b

40 1.5. BRAIDINGS ON THE CATEGORY OF BIMODULES

1 1 R1 ⊗ i R2 ⊗ R3 = 1 ⊗ i ⊗ 1 + 1 ⊗ 1 ⊗ i + i ⊗ 1 ⊗ 1) + i ⊗ k ⊗ k − k ⊗ i ⊗ k 4 4ab 1 − k ⊗ k ⊗ i+ 1 ⊗ k ⊗ j − 1 ⊗ j ⊗ k − i ⊗ j ⊗ j + j ⊗ i ⊗ j + j ⊗ k ⊗ 1 4b 1 − j ⊗ 1 ⊗ k − k ⊗ j ⊗ 1 + k ⊗ 1 ⊗ j + j ⊗ j ⊗ i+ i ⊗ i ⊗ i 4a

1 1 R1 ⊗ R2 ⊗ R3 i = 1 ⊗ 1 ⊗ i + 1 ⊗ i ⊗ 1 + i ⊗ 1 ⊗ 1+ i ⊗ k ⊗ k − k ⊗ k ⊗ i 4 4ab 1 − k ⊗ i ⊗ k+ 1 ⊗ k ⊗ j − 1 ⊗ j ⊗ k − i ⊗ j ⊗ j − j ⊗ 1 ⊗ k + j ⊗ j ⊗ i 4b 1 + j ⊗ i ⊗ j + j ⊗ k ⊗ 1 + k ⊗ 1 ⊗ j − k ⊗ j ⊗ 1+ i ⊗ i ⊗ i 4a

Thus R1 ⊗ i R2 ⊗ R3 = R1 ⊗ R2 ⊗ R3 i. By a straightforward computation it can be seen that we also have R1 ⊗ j R2 ⊗ R3 = R1 ⊗ R2 ⊗ R3 j and R1 ⊗ k R2 ⊗ R3 = R1 ⊗ R2 ⊗ R3 k.

Example 1.5.11 If A is an Azumaya algebra, then there exists an R such that (Ae,R) is quasitriangular. The converse is not true, it sufﬁces to consider Q as a Z-algebra. Since Z ⊂ Q is an epimorphism of rings, it follows from Proposition 1.5.3 that (Q ⊗Z Q = Q, 1 ⊗ 1 ⊗ 1) is quasitriangular; it is obvious that Q is not a Z-Azumaya algebra.

Example 1.5.12 Let A and B be k-algebras, and assume that (Ae,R) and (Be,S) are quasitriangular. It is straightforward to show that ((A ⊗ B)e,T ), with T = R1 ⊗ S1 ⊗ R2 ⊗ S2 ⊗ R3 ⊗ S3 ∈ (A ⊗ B)(3), is quasitriangular.

We conclude this Section with another application of canonical R-matrices: they can be applied to deform the switch map into a simultaneous solution of the quantum Yang-Baxter equation and the braid equation.

Theorem 1.5.13 Let (A, R) be an algebra with a canonical R-matrix and V an A-bimodule. Then the map Ω: V ⊗ V → V ⊗ V, Ω(v ⊗ w) = R1 wR2 ⊗ R3 v is a solution of the quantum Yang-Baxter equation Ω12 Ω13 Ω23 = Ω23 Ω13 Ω12 and of the braid equation Ω12 Ω23 Ω12 = Ω23 Ω12 Ω23. Moreover Ω3 = Ω in End(V (3)).

Proof: Let r = S = R. Then for any v, w, t ∈ V we have:

Ω12 Ω13 Ω23(v ⊗ w ⊗ t) = Ω12 Ω13(v ⊗ R1 t R2 ⊗ R3 w) = Ω12(r1 R3 w r2 ⊗ R1 t R2 ⊗ r3 v) = S1 R1 t R2 S2 ⊗ S3 r1 R3 w r2 ⊗ r3 v (1.26) (1.32) = R1 t R2 S1 S2 ⊗ S3 r1 R3 w r2 ⊗ r3 v = R1 t R2 ⊗ r1 R3 w r2 ⊗ r3 v (1.25) = R1 t R2 ⊗ r1 w r2 ⊗ R3 r3 v

41 CHAPTER 1. CATEGORICAL CONSTRUCTIONS and

Ω23 Ω13 Ω12(v ⊗ w ⊗ t) = Ω23 Ω13(R1 w R2 ⊗ R3 v ⊗ t) = Ω23(r1 t r2 ⊗ R3 v ⊗ r3 R1 w R)2) = r1 t r2 ⊗ S1 r3 R1 w R2 S2 ⊗ S3 R3 v (1.25) (1.26) = r1 t r2 ⊗ S1 R1 w R2 S2 ⊗ r3 S3 R3 v = r1 t r2 ⊗ R1 w R2 S1 S2 ⊗ r3 S3 R3 v (1.32) = r1 t r2 ⊗ R1 w R2 ⊗ r3 R3 v,

Hence Ω is a solution of the quantum Yang-Baxter equation. On the other hand:

Ω12 Ω23 Ω12(v ⊗ w ⊗ t) = Ω12 Ω23(R1 w R2 ⊗ R3 v ⊗ t) = Ω12(R1 w R2 ⊗ r1 t r2 ⊗ r3 R3v) = S1 r1 t r2 S2 ⊗ S3 R1 w R2 ⊗ r3 R3 v (1.26) (1.32) = r1 t r2 S1 S2 ⊗ S3 R1 w R2 ⊗ r3 R3 v = r1 t r2 ⊗ R1 w R2 ⊗ r3 R3 v and

Ω23 Ω12 Ω23(v ⊗ w ⊗ t) = Ω23 Ω12(v ⊗ r1 t r2 ⊗ r3 w) = Ω23(S1 r1 t r2 S2 ⊗ S3 v ⊗ r3 w) = S1 r1 t r2 S2 ⊗ R1 r3 w R2 ⊗ R3 S3v (1.26) (1.32) = r1 t r2 S1 S2 ⊗ R1 r3 w R2 ⊗ R3 S3v = r1 t r2 ⊗ R1 r3 w R2 ⊗ R3 v (1.25) = r1 t r2 ⊗ R1 w R2 ⊗ r3 R3 v.

Thus Ω is also a solution of the braid equation. Finally,

Ω3(v ⊗ w) = S1 r3 R1 w R2 S2 ⊗ S3 r1 R3 v r2 (1.25) (1.34) = S1 w R2 S2 ⊗ r3 R1 S3 r1 R3 v r2 = S1 w S2 ⊗ r3 R1 S3 R2 r1 R3 v r2 (1.26) (1.32) = S1 w S2 ⊗ R1 S3 R2 r3 r1 R3 v r2 = S1 w S2 ⊗ R1 S3 R2 R3 v (1.33) = S1 w S2 ⊗ S3 v = Ω(v ⊗ w).

42 1.6. THE CENTER OF THE CATEGORY OF BIMODULES

1.6 The center of the category of bimodules

A monoidal category can be viewed as a categorical version of a monoid. The appropriate generalization of the center of a monoid is given by the centre construction, which was introduced independently by Drinfeld (unpublished), Joyal and Street [80] and Majid [98]. A key result in the classical theory is the following: the center of the category of representations of a Hopf algebra H is isomorphic to the category of Yetter-Drinfeld modules over H [81]. Moreover, if the Hopf algebra H is finite diminesional, then the category of Yetter-Drinfeld modules is isomorphic to the category of representations over the Drinfeld double D(H). Since the center is a braided monoidal category, it follows that the Drinfeld double is a quasitriangular Hopf algebra. Let A be an algebra over a commutative ring k. In this section, we study the center of the category op AMA of A-bimodules, and relate it to some classical concepts. We introduce A ⊗ A -Yetter- Drinfeld modules (Definition 1.6.1), and show that the weak center of AMA is isomorphic to the category of A ⊗ Aop-Yetter-Drinfeld modules (Proposition 1.6.3). We give other descriptions: the weak center is equal to the center (Proposition 1.6.5) and is isomorphic to the category MA⊗A of comodules over the Sweedler canonical coring A ⊗ A (Proposition 1.6.2). Moreover it was proved in [41, Theorem 5] that the category MA⊗A is isomorphic to the category of right A-modules with a flat connection as defined in noncommutative differential geometry. Thus, the category of right A-modules with a flat connection is also equal to the center. We introduce a category of descent data Desc(A/k), generalizing the descent data introduced in [83] from A commutative to A non-commutative, and this category is also isomorphic to the center. The first main result of this section is summarized in Theorem 1.6.9 which provide six isomorphic descriptions for the center of the category of A-bimodules. All six isomorphic categories are braided monoidal categories. In particular, the category of comodules over the Sweedler canonical A-coring A ⊗ A is a braided monoidal category and hence one can perform most of the constructions that are performed for differentiable manifolds. For instance, connections in bimodules try to mimic linear connections in geometry and are useful in capturing Riemannian aspects (see [43], [42] for more detalis). If A is faithfully flat as a k-module, all these categories are equivalent to the category of k-modules, by classical descent theory. In the case where A is finitely generated and projective, the category MA⊗A is isomorphic to the category of left modules over Endk(A), in fact, one may view Endk(A) as the Drinfeld double of the enveloping e op algebra A = A ⊗ A . Thus, the category of left Endk(A)-modules is braided monoidal, and we give an explicit description of the monoidal structure and the braiding in terms of the finite dual basis of A. If we apply this to the case where A = kn, then we find that the category of left modules over the matrix ring Mn(k) is braided monoidal. We give an explicit description of the tensor product and the braiding. The second major application of the above results is the fact that they lead to constructing new and interesting family of solutions for the quantum Yang-Baxter equation. If V is a right comodule over the Sweedler canonical coring A ⊗ A, then the canonical map Ω: V ⊗ V → V ⊗ V , 3 Ω(v ⊗ w) = w[0] ⊗ v[0]w[1]v[1] is a solution of the quantum Yang-Baxter equation and Ω = Ω in the endomorphism algebra End(V ⊗ V ) (Theorem 1.6.14). Several examples are provided.

43 CHAPTER 1. CATEGORICAL CONSTRUCTIONS

Deﬁnition 1.6.1 A right Yetter-Drinfeld Ae-module consists of a pair (V, ρ), such that V is an A-bimodule, (V, ρ) ∈ MA⊗A and the following compatibility conditions hold:

ρ(av) = v[0] ⊗ av[1]; (1.36)

aρ(v) = av[0] ⊗ v[1] = v[0]a ⊗ v[1]. (1.37)

A morphism (V, ρ) → (V 0, ρ0) of Yetter-Drinfeld modules is a map f : V → V 0 that is an A-bimodule and A(2)-comodule map. The category of Yetter-Drinfeld modules will be denoted e by YDA .

e Take (V, ρ) ∈ YDA . Then

(1.7) (1.36) av = (av)[0](av)[1] = v[0]av[1], (1.38) and (1.38) (1.8) (1.7) v[1]v[0] = v[0][0]v[1]v[0][1] = v[0]v[1] = v. (1.39)

e Proposition 1.6.2 The forgetful functor U : YDA → MA⊗A is an isomorphism of categories.

e Proof: We define a functor P : MA⊗A → YDA . For V ∈ MA⊗A, let P (V ) = V as an (2) A -comodule, with left A-action defined by av = v[0]av[1]. Then (1.9) (1.8) ρ(av) = ρ(v[0]av[1]) = ρ(v[0])av[1] = v[0] ⊗ av[1], and (1.36) is satisfied. The left A-action is associative since

(1.36) b(av) = (av)[0]b(av)[1] = v[0]bav[1] = (ba)v.

Finally we show that (1.37) holds:

(1.8) av[0] ⊗ v[1] = v[0][0]av[0][1] ⊗ v[1] = v[0]a ⊗ v[1].

e This shows that P (V ) ∈ YDA . If f : V → W is a morphism in MA⊗A, then it is also a e morphism P (V ) → P (W ) in YDA . To this end, we need to show that f is left A-linear:

f(av) = f(v[0]av[1]) = f(v[0])av[1] = f(v)[0]af(v)[1] = af(v).

We used the fact that f is right A-linear and right A ⊗ A-colinear. Finally, it is clear that the functors P and V are inverses.

Recall from Section 1.1 that Wr(AMA) is the weak right center of the monoidal category (AMA, − ⊗A −,A) of A-bimodules.

44 1.6. THE CENTER OF THE CATEGORY OF BIMODULES

Ae Proposition 1.6.3 The categories Wr(AMA) and YD are isomorphic.

Proof: Let (V, c−,V ) be an object of Wr(AMA). For every A-bimodule M, we have an A- bimodule map cM,V : M ⊗A V → V ⊗A M, which is natural in M. Consider

(2) ∼ (2) ∼ g = cA⊗A,V : A ⊗A V = A ⊗ V → V ⊗A A = V ⊗ A, and deﬁne ρ : V → V ⊗ A by ρ(v) = g(1 ⊗ v) = v[0] ⊗ v[1] ∈ V ⊗ A. c−,V is then completely (2) determined by ρ: for m ∈ M, deﬁne the A-bimodule map fm : A → M by the formula fm(a ⊗ b) = amb. From the naturality of c−,V , it follows that we have a commutative diagram

g (2) (2) A ⊗A V / V ⊗A A

fm⊗AV V ⊗Afm cM,V M ⊗A V / V ⊗A M

Evaluating the diagram at 1 ⊗ v, we ﬁnd

cM,V (m ⊗A v) = v[0] ⊗A m v[1]. (1.40)

Ae We will now show that (V, ρ) ∈ YD . Using the fact that cM,V is right A-linear, well-deﬁned and left A-linear, we ﬁnd

(va)[0] ⊗ m(va)[1] = cM,V (m ⊗A va) = cM,V (m ⊗A v)a = v[0] ⊗A mv[1]a;

v[0] ⊗A mav[1] = cM,V (ma ⊗A v) = cM,V (m ⊗A av) = (av)[0] ⊗ m(av)[1];

v[0] ⊗A amv[1] = cM,V (am ⊗A v) = acM,V (m ⊗A v) = av[0] ⊗A mv[1].

(2) If we take M = A and m = 1 ⊗ 1 in these formulas, we obtain (1.9), (1.36) and (1.37). cA,V is the canonical isomorphism A ⊗A V → V ⊗A A, hence v ⊗A 1 = cA,V (1 ⊗A v) = v[0] ⊗A v[1], and (1.7) follows. Finally, we have the commutative diagram

cM⊗AN,V M ⊗A N ⊗A V / V ⊗A M ⊗A N SSS kk5 SSS kkk SSS kkk M⊗AcN,VSSS kkckM,V ⊗AN SS) kkk M ⊗A V ⊗A N

We evaluate the diagram at m ⊗A n ⊗A v:

v[0] ⊗A m ⊗A nv[1] = cM⊗AN,V (m ⊗A n ⊗A v) = (cM,V ⊗A N) ◦ (M ⊗A cN,V ) (m ⊗A n ⊗A v)

= (cM,V ⊗A N)(m ⊗A v[0] ⊗A nv[1]) = v[0][0] ⊗A mv[0][1] ⊗A ⊗Anv[1]

(1.8) follows after we take M = N = A(2) and m = n = 1 ⊗ 1. Ae Conversely, given (V, ρ) ∈ YD , we deﬁne c−,V using (1.40). Straightforward computations show that (V, c−,V ) ∈ Wr(AMA).

45 CHAPTER 1. CATEGORICAL CONSTRUCTIONS

Our next aim is to show that condition (1.7) in Deﬁnition 1.6.1 can be replaced by the condition that g is invertible.

Proposition 1.6.4 Let A be a k-algebra, and assume that ρ : V → V ⊗ A satisﬁes (1.8)-(1.9) and (1.36)-(1.37). Then (1.7) holds if and only if g : A ⊗ V → V ⊗ A, g(a ⊗ v) = av[0] ⊗ v[1] is invertible.

Proof: Assume that (1.7) holds. For all a ∈ A and v ∈ V , we have

(τ ◦ g ◦ τ ◦ g)(a ⊗ v) = (τ ◦ g)(v[1] ⊗ av[0]) (1.36) = τ v[1](av[0])[0] ⊗ (av[0])[1] = τ v[1]v[0][0] ⊗ av[0][1] (1.37) (1.8) (1.7) = τ v[0][0]v[1] ⊗ av[0][1] = τ(v[0]v[1] ⊗ a) = a ⊗ v.

We conclude that τ ◦ g ◦ τ ◦ g = IdA⊗V . Composing to the left and to the right with the switch −1 map τ, we ﬁnd g ◦ τ ◦ g ◦ τ = IdV ⊗A. Thus g = τ ◦ g ◦ τ. Conversely, assume that g is invertible. For any v ∈ V we have: (1.9) (1.8) g(1 ⊗ v[0]v[1]) = ρ(v[0]v[1]) = ρ(v[0])v[1] = v[0] ⊗ v[1] = g(1 ⊗ v).

−1 (1.7) follows after we apply g to both sides and multiply the two tensor factors.

Proposition 1.6.5 The (right) center of the category of A-bimodules coincides with its (right) weak center: Zr(AMA) = Wr(AMA).

Proof: Take (V, c−,V ) ∈ Wr(AMA). We will show that cM,V is invertible, for every A- bimodule M. Let g and ρ be as in Proposition 1.6.3. We claim that

−1 cM,V (v ⊗A m) = v[1]m ⊗A v[0]. (1.41) Indeed, for all m ∈ M and v ∈ V , we have that

−1 (1.40),(1.41) (1.8) (1.39) (cM,V ◦ cM,V )(m ⊗A v) = v[0][1]mv[0][0] ⊗A v[0] = m ⊗A v[1]v[0] = m ⊗A v;

−1 (1.41),(1.40) (1.8) (1.7) (cM,V ◦ cM,V )(v ⊗A m) = v[0][0] ⊗A v[1]mv[0][1] = v[0]v[1] ⊗A m = v ⊗A m.

If V and W are A-bimodules, then V ⊗ W is an A(2)-bimodule. Consider a map g : A ⊗ V → V ⊗ A in A(2) MA(2) . The maps g1, g2, g3 deﬁned by (1.6) are in A(3) MA(3) .

Deﬁnition 1.6.6 Let A be a k-algebra. A descent datum consists of an A-bimodule V together (2) with an A -bimodule map g : A ⊗ V → V ⊗ A such that g2 = g3 ◦ g1 and (ψ ◦ g)(a ⊗ v) = v, for all v ∈ V , where ψ is the map V ⊗A → A, ψ(v⊗a) = va. A morphism between two descent data (V, g) and (V 0, g0) is an A-bimodule map f : V → V 0 such that (f ⊗A)◦g = g0 ◦(A⊗f). The category of descent data is denoted by Desc(A/k).

46 1.6. THE CENTER OF THE CATEGORY OF BIMODULES

e Proposition 1.6.7 The categories Desc(A/k) and YDA are isomorphic.

Ae Proof: Let (V, ρ) ∈ YD , and deﬁne g : A ⊗ V → V ⊗ A by g(a ⊗ v) = av[0] ⊗ v[1]. First we show that g is an A(2)-bimodule map. (1.36) g(ba ⊗ cv) = ba(cv)[0] ⊗ (cv)[1] = bav[0] ⊗ cv[1] = (b ⊗ c)g(a ⊗ v); (1.9) (1.37) g(ab ⊗ vc) = ab(vc)[0] ⊗ (vc)[1] = abv[0] ⊗ v[1]c = av[0]b ⊗ v[1]c = g(a ⊗ v)(b ⊗ c).

Now g3 ◦ g1 = g2 since

(g3 ◦ g1)(a ⊗ b ⊗ v) = g3(a ⊗ bv[0] ⊗ v[1]) = a(bv[0])[0] ⊗ (bv[0])[1] ⊗ v[1] (1.36) (1.8) = av[0][0] ⊗ bv[0][1] ⊗ v[1] = av[0] ⊗ b ⊗ v[1] = g2(a ⊗ b ⊗ v).

Finally, (m ◦ g)(1 ⊗ v) = v[0]v[1] = v, and we conclude that (V, g) ∈ Desc(A/k). Conversely, let (V, g) ∈ Desc(A/k), and deﬁne ρ : V → V ⊗ A by ρ(v) = g(1 ⊗ v). Then f(a ⊗ v) = aρ(v) = av[0] ⊗ v[1]. It is easy to show that (1.7), (1.9), (1.36) and (1.37) are satisﬁed:

v = (m ◦ g)(1 ⊗ v) = m(ρ(v)) = v[0]v[1];

ρ(va) = g(1 ⊗ va) = g(1 ⊗ v)(1 ⊗ a) = v[0] ⊗ v[1]a;

ρ(av) = g(1 ⊗ av) = (1 ⊗ a)g(1 ⊗ v) = v[0] ⊗ av[1];

aρ(v) = (a ⊗ 1)g(v) = g(a ⊗ v) = g(1 ⊗ v)(a ⊗ 1) = v[0]a ⊗ v[1].

We have already computed g3 ◦ g1 and g2. This computation stays valid, since we only used (1.36), which holds. Expressing that (g3 ◦ g1)(1 ⊗ 1 ⊗ v) = g2(1 ⊗ 1 ⊗ v), we ﬁnd (1.8). We Ae conclude that (V, ρ) ∈ YD .

Remarks 1.6.8 1. It follows from the proof of Proposition 1.6.7 that the deﬁnition of a descent datum can be restated as follows: V ∈ AMA, an invertible map g : A ⊗ V → V ⊗ A in A(2) MA(2) satisfying g2 = g3 ◦ g1. 2. We look at the particular case where A is commutative. Take (V, g) ∈ Desc(A/k) and let Ae (1.38) (1.7) (V, ρ) be the corresponding object of YD . Then we know that av = v[0]av[1] = v[0]v[1]a = va, hence the left A-action on V coincides with the right A-action. Consequently, the left and right A(2)-actions on A ⊗ V and V ⊗ A coincide. So we can view a descent datum (V, g) as a right (2) A-module V together with a right A -linear map g : A ⊗ V → V ⊗ A satisfying g3 = g3 ◦ g1 and (ψ ◦ g)(1 ⊗ v) = v, or, equivalently, g invertible. These are precisely the descent data [83] that we discussed in Section 1.1.

The main results of this section are summarized as follows:

Ae A⊗A Theorem 1.6.9 For a k-algebra A, the categories Desc(A/k), YD , M , Wr(AMA) and Zr(AMA) are isomorphic. If A is faithfully ﬂat over k then these isomorphic categories are equivalent to the category of k-modules.1

1 The fact that Zr(AMA) is equivalent to the category of k-modules if A is faithfully ﬂat can be also derived from [135, Theorem 3.3].

47 CHAPTER 1. CATEGORICAL CONSTRUCTIONS

Zr(AMA) is a braided monoidal category, hence we can deﬁne braided monoidal structures on the ﬁve isomorphic categories in Theorem 1.6.9. In particular, the category of comodules over the Sweedler canonical A-coring A ⊗ A is braided monoidal. Explicitly we have:

A⊗A Corollary 1.6.10 Let A be a k-algebra. Then (M , − ⊗A −,A) is a braided monoidal A⊗A category as follows: for V ∈ M , we have a left A-action on V deﬁned by a · v = v[0]av[1]. 0 The tensor product is then just the tensor product over A, and the coaction on V ⊗AV is given by 0 0 0 the formula ρ(v ⊗A v ) = v[0] ⊗A v[0] ⊗v[1]v[1]. The unit is A, with A⊗A-coaction ρ(a) = 1⊗a. The left A-action on A then coincides with the left regular representation: b · a = a[0]ba[1] = ba. The braiding c on MA⊗A is given by

0 0 −1 0 0 cV 0,V (v ⊗A v) = v[0] ⊗A v v[1] ; cV 0,V (v ⊗A v ) = v[1]v ⊗A v[0].

Proof: This follows of course from the general theory of the center construction, but all axioms can be easily veriﬁed directly.

Remark 1.6.11 An interesting interpretation of Theorem 1.6.9 and Corollary 1.6.10 was com- municated to us by T. Brzezinski. In [41], it was observed that there is a close relationship between corings with a grouplike element and noncommutative differential geometry. One of the results in this direction is the following: the category MA⊗A is isomorphic to the category Conn(A/k, Omega(A ⊗ A/k) of right A-modules with a ﬂat connection, see [41, Theorem 5] or [43, Sec. 29]. It then follows from Corollary 1.6.10 that Conn(A/k, Omega(A ⊗ A/k) is a braided monoidal category. In the forthcoming [42], the braiding on MA⊗A is applied to prove that any ﬂat connection in a right A-module is an A-bimodule connection.

Finitely generated projective algebras

Now we focus attention to the case where A is ﬁnitely generated and projective as a k-module, which means that the k-linear map

∗ ∗ ∗ ϕ : A ⊗ A → A = Endk(A), ϕ(a ⊗ b)(x) = ha , xib (1.42)

−1 P ∗ is an isomorphism. Then ϕ (IdA) = i ai ⊗ ai is called a ﬁnite dual basis of A, and is P ∗ characterized by the formula ihai , xiai = x, for all x ∈ A. In this situation, we also have that

−1 X ∗ ϕ (f) = ai ⊗ f(ai), (1.43) i

P ∗ P ∗ for all f ∈ A. Indeed, ϕ( i ai ⊗ f(ai))(x) = ihai , xif(ai) = f(x), for all x ∈ A. Recall op that we also have an algebra map F : A ⊗ A → Endk(A), F (a ⊗ b)(x) = axb. It is then easy to show that

ϕ(a∗ ⊗ a) = F (a ⊗ 1) ◦ ϕ(a∗ ⊗ 1) = F (1 ⊗ a) ◦ ϕ(a∗ ⊗ 1). (1.44)

48 1.6. THE CENTER OF THE CATEGORY OF BIMODULES

A⊗A The categories M and AM are isomorphic. If V is a right A ⊗ A-comodule, then we have a left A-action given by

f · v = v[0]f(v[1]). (1.45) for all f ∈ A = Endk(A) and v ∈ V . Conversely, for V ∈ AM, we have a right A⊗A-coaction now given by X ρ(v) = fi · v ⊗ ai, (1.46) i ∗ where we write fi = ϕ(ai ⊗ 1). This is well-known and can be veriﬁed easily. It also has an ∼ explanation in terms of corings: the left dual of the A-coring A ⊗ A is AHom(A ⊗ A, A) = End(A)op as A-rings, see for example [43]. We will now transport the braided monoidal struc- A⊗A ture of M to AM.

If V ∈ AM, then V ∈ AMA, by restriction of scalars via F . Now we also have that V ∈ e MA⊗A =∼ YDA , and this gives a second A-bimodule structure on V . These two bimodule structures coincide:

(1.45) F (1 ⊗ a) · v = v[0](F (1 ⊗ a)(v[1])) = v[0]v[1]a = va; (1.45) F (a ⊗ 1) · v = v[0](F (a ⊗ 1)(v[1])) = v[0]av[1] = av.

A⊗A ∼ Now take V , W ∈ AM. Then V ⊗A W ∈ M = AM. We describe the A-action on V ⊗A W . (1.45) (1.46) X f · (v ⊗A w) = v[0] ⊗A w[0]f(v[1]w[1]) = fi · v ⊗A (fj · w)f(aiaj) i,j X ∗ = fi · v ⊗A F (1 ⊗ f(aiaj)) ◦ ϕ(aj ⊗ 1) · w i,j (1.44) X ∗ (1.43) X = fi · v ⊗A ϕ(aj ⊗ f(aiaj)) · w = fi · v ⊗A f(ai−) · w, i,j i where f(a−) ∈ A is the map sending x ∈ A to f(ax); we have an alternative description:

X ∗ f · (v ⊗A w) = fi · v ⊗A F (1 ⊗ f(aiaj)) ◦ ϕ(aj ⊗ 1) · w i,j (1.44) X ∗ = fi · v ⊗A F (f(aiaj) ⊗ 1) ◦ ϕ(aj ⊗ 1) · w i,j X ∗ = F (1 ⊗ f(aiaj)) ◦ ϕ(ai ⊗ 1) · v ⊗A fj · w i,j (1.44) ∗ (1.43) X = ϕ(ai ⊗ f(aiaj)) · v ⊗A fj · w = f(−aj) · v ⊗A fj · w. j P The braiding is given by the formula cV,W (v ⊗A w) = w[0] ⊗A vw[1] = i fi · w ⊗ vai. We summarize our results:

49 CHAPTER 1. CATEGORICAL CONSTRUCTIONS

Proposition 1.6.12 Let A be a ﬁnitely generated projective k-algebra, with ﬁnite dual basis P ∗ ∗ i ai ⊗ ai, and write fi = ϕ(ai ⊗ 1). The category of left Endk(A)-modules is a braided monoidal category. The tensor product is the tensor product over A; a left Endk(A)-module is an A-bimodule by restriction of scalars via F . The left Endk(A)-action on V ⊗A W is given by X X f · (v ⊗A w) = fi · v ⊗A f(ai−) · w = f(−aj) · v ⊗A fj · w i j for all f ∈ Endk(A), v ∈ V and w ∈ W . The unit object is A, with its obvious left Endk(A)- P action f · a = f(a). The braiding is given by cV,W (v ⊗A w) = i fi · w ⊗A v ai.

n n Example 1.6.13 Let A = k = ⊕i=1kei, with multiplication eiej = δijei and unit 1 = Pn ∗ ∗ ∗ i=1 ei. Let ei ∈ A be given by hei , eji = δij. We can then identify Mn(k) and Endk(A), where an endomorphism of A corresponds to its matrix with respect to the basis {e1, ··· , en}. ∗ It is then easy to see that ϕ(ei ⊗ ej) = eji, the elementary matrix with 1 in the (i, j)-position P ∗ P and 0 elsewhere. Now we easily compute that fl = ϕ r el ⊗er = r erl, eii = F (ei ⊗1) = F (1⊗ei) and eij(el−) = δjleij. Let V and W be left Mn(k)-modules. Then V ⊗kn W is again a left Mn(k)-module, the left Mn(k)-action is given by the formulas in Proposition 1.6.12, which simplify as follows: X X eij · (v ⊗kn w) = erl · v ⊗kn δjleij · w = erj · v ⊗kn eij · w l,r r X X = erj · v ⊗kn (eiieij) · w = erj · v ⊗kn ei(eij · w) r r X X = (erj · v)ei ⊗kn eij · w = (eiierj) · v ⊗kn eij · w r r = eij · v ⊗kn eij · w.

Finally, we compute the braiding X X cV,W (v ⊗kn w) = fi · w ⊗kn vei = eri · w ⊗kn eiv i i,r X X = (eri · w)ei ⊗kn v = (eiieri) · w ⊗kn v i,r i,r X = eii · w ⊗kn v = w ⊗kn v. i

New solutions for the quantum Yang-Baxter equation

Our results lead to the construction of a new family of solutions of the quantum Yang-Baxter e equation. More precisely, to every object of YDA =∼ MA⊗A, we can associate a solution of the quantum Yang-Baxter equation.

50 1.6. THE CENTER OF THE CATEGORY OF BIMODULES

e Theorem 1.6.14 Let A be a k-algebra and (V, ρ) ∈ YDA . Then

Ω = Ω(V,ρ) : V ⊗ V → V ⊗ V, Ω(v ⊗ w) = w[0] ⊗ w[1]v, (1.47) is a solution of the quantum Yang-Baxter equation Ω12 Ω13 Ω23 = Ω23 Ω13 Ω12 in End(V (3)). In particular, if (V, ρ) ∈ MA⊗A, then

Ω = Ω(V,ρ) : V ⊗ V → V ⊗ V, Ω(v ⊗ w) = w[0] ⊗ v[0]w[1]v[1], (1.48) is a solution of the quantum Yang-Baxter equation and Ω3 = Ω in End(V (2)).

Proof: For all v, w, t ∈ V , we have that:

12 13 23 12 13 Ω Ω Ω (v ⊗ w ⊗ t) = Ω Ω (v ⊗ t[0] ⊗ t[1] w) 12 = Ω (t[1]w)[0] ⊗ t[0] ⊗ (t[1]w)[1] v = t[0][0] ⊗ t[0][1] (t[1]w)[0] ⊗ (t[1]w)[1] v (1.36) (1.8) = t[0][0] ⊗ t[0][1] t[1] w[0] ⊗ w[1] v = t[0] ⊗ t[1] w[0] ⊗ w[1] v; 23 13 12 23 13 Ω Ω Ω (v ⊗ w ⊗ t) = Ω Ω (w[0] ⊗ w[1] v ⊗ t) 23 = Ω (t[0] ⊗ w[1] v ⊗ t[1] w[0]) = t[0] ⊗ (t[1]w[0])[0] ⊗ (t[1]w[0])[1] w[1] v (1.36) (1.8) = t[0] ⊗ t[1] w[0][0] ⊗ w[0][1] w[1] v = t[0] ⊗ t[1] w[0] ⊗ w[1] v

Thus Ω12 Ω13 Ω23 = Ω23 Ω13 Ω12. We have seen in Proposition 1.6.2 that (V, ρ) ∈ MA⊗A, with Ae left A-action a · v = v[0]av[1], is an object of YD . With this identiﬁcation the canonical map (1.47) takes precisely the form (1.48). Now, for all v, w ∈ V we have:

2 (1.9) Ω (v ⊗ w) = Ω(w[0] ⊗ v[0]w[1]v[1]) = v[0][0] ⊗ w[0][0]v[0][1]w[1]v[1]w[0][1] (1.8) (1.7) = v[0] ⊗ w[0]w[1]v[1] = v[0] ⊗ w v[1]; 3 (1.9) Ω (v ⊗ w) = Ω(v[0] ⊗ w v[1]) = w[0] ⊗ v[0][0]w[1]v[1]v[0][1] (1.8) = w[0] ⊗ v[0]w[1]v[1] = Ω(v ⊗ w).

It is well-known that x = x1 ⊗ x2 ∈ A(2) is grouplike if and only if x1x2 = 1 and X1 ⊗ X2x1 ⊗ x2 = X1 ⊗ 1 ⊗ X2. Grouplike elements of a coring C are in bijective correspondence to right C-coactions on A. In the case where C = A(2), the right A ⊗ A-coaction on A associated to x is ρ(a) = x1 ⊗ x2a. For M ∈ MA⊗A, we deﬁne

M cox = {m ∈ M | ρ(m) = mx1 ⊗ x2}.

Acox is a subalgebra of A. Suppose that we have an algebra morphism i : B → Acox. Then cox we have a pair of adjoint functors (see [46, Sec. 1]) (− ⊗B A, (−) ) between the categories A⊗A 1 2 MB and M . The right coaction on N ⊗B A is simply ρ(n ⊗B a) = n ⊗B x ⊗ x a. This construction allows us to give examples of A ⊗ A-comodules, and, a fortiori, solutions of the quantum Yang-Baxter equation, applying Theorem 1.6.14. We then obtain the following.

51 CHAPTER 1. CATEGORICAL CONSTRUCTIONS

Proposition 1.6.15 Let x be a grouplike element of A ⊗ A, and let i : B → Acox be an algebra (2) (2) morphism. For N ∈ MB, the map Ω:(N ⊗B A) → (N ⊗B A) given by

1 1 2 2 Ω((n ⊗B a) ⊗ (m ⊗B b)) = (m ⊗B x ) ⊗ (n ⊗B X x bX a) (1.49) is a solution of the quantum Yang-Baxter equation.

As a particular example, we can take x = 1 ⊗ 1, B = k, N ∈ Mk. Then (1.49) takes the form Ω(m ⊗ a ⊗ n ⊗ b) = n ⊗ 1 ⊗ m ⊗ b a. In particular, if we take N = k, then Ω(a ⊗ b) = 1 ⊗ ba, and this shows that Ω is not necessarily bijective.

We now present another way to construct comodules over A ⊗ A. Yetter-Drinfeld modules can be constructed from bimodules over quasitriangular algebras as follows.

Proposition 1.6.16 Let A be a k-algebra, let V be an A-bimodule, and let R ∈ A(3) be an R- 1 2 3 Ae matrix. Consider ρR : V → V ⊗A, ρR(v) = R v R ⊗R = v[0] ⊗v[1]. Then (V, ρR) ∈ YD , and the associated solution of the quantum Yang-Baxter equation is

1 2 3 ΩR = Ω(V,ρR) : V ⊗ V → V ⊗ V, ΩR(v ⊗ w) = R wR ⊗ R v.

A⊗A Proof: We show that (V, ρR) ∈ M , that is, ρ satisﬁes (1.7)-(1.8). (1.7) follows from (1.32). (1.8) is equivalent to

1 2 1 2 3 1 2 3 (R v R )[0] ⊗ (R v R )[1] ⊗ R = R v R ⊗ 1 ⊗ R and to r1 R1 v R2 r2 ⊗ r3 ⊗ R3 = R1 v R2 ⊗ 1 ⊗ R3. (1.50) Using (1.27)-(1.28), we obtain:

(1.28) (1.27) R1 ⊗ R2 ⊗ 1 ⊗ R3 = r1 R1 ⊗ r2 ⊗ r3 R2 ⊗ R3 = r1 R1 ⊗ R2 r2 ⊗ r3 ⊗ R3

e and (1.50) follows. It follows from Proposition 1.6.2 that (V, ρ) ∈ YDA , and we are done if we can show that the left A-action on V given by (1.38) coincides with the original left A-action. This can be shown easily:

1 2 3 1 2 3 v0]av[1] = R vR aR = aR vR R = av.

We used (1.27)-(1.32), combined with the fact that R is invariant under cyclic permuation of the tensor factors.

Several examples of R-matrices are presented in [10]. In particular, if A is an Azumaya algebra, then we have a unique R-matrix. Applying Proposition 1.6.16 to [10, Example 2.8], we obtain the following.

52 1.6. THE CENTER OF THE CATEGORY OF BIMODULES

Example 1.6.17 Let A = Mn(k) be a matrix algebra and V an Mn(k)-bimodule. Then the map n X Ω: V ⊗ V → V ⊗ V, Ω(v ⊗ w) = eij w eki ⊗ ejk v i,j,k=1 is a solution of the quantum Yang-Baxter equation. eij is the elementary matrix with 1 in the (i, j)-position and 0 elsewhere.

Bibliographical Notes

The material in Section 1.1 contains classical well-known results that can be found, for instance, in [1], [2], [43], [47], [61], [69], [81], [83], [93], [139]. The results of Section 1.2, Section 1.3 and Section 1.4 are taken from the author’s papers [3], [4] and [5]. The results of Section 1.5 and Section 1.6 are part of the author’s joint papers with S. Caenepeel and G. Militaru [10] and [9].

53 CHAPTER 1. CATEGORICAL CONSTRUCTIONS

54 2 Uniﬁed products

2.1 Notational conventions

Throughout this chapter, k will be a ﬁeld. Unless speciﬁed otherwise, all algebras, coalgebras, bialgebras, tensor products and homomorphisms are over k. For a coalgebra C, we use Sweedler’s Σ-notation: ∆(c) = c(1) ⊗ c(2), (I ⊗ ∆)∆(c) = c(1) ⊗ c(2) ⊗ c(3), etc (summation understood). We also use the Sweedler notation for left C-comodules: ρ(m) = m<−1> ⊗m<0>, for any m ∈ M if (M, ρ) ∈ C M is a left C-comodule. Let A be a bialgebra and H a coalgebra. H is called a right A-module coalgebra if there exists / : H ⊗ A → H a morphism of coalgebras such that (H,/) is a right A-module. For a k-linear map f : H ⊗ H → A we denote f(g, h) = f(g ⊗ h); f is the trivial map if f(g, h) = εH (g)εH (h)1A, for all g, h ∈ H. Simi- larly, the k-linear maps / : H ⊗A → H, . : H ⊗A → A are the trivial actions if h/a = εA(a)h and respectively h . a = εH (h)a, for all a ∈ A and h ∈ H. For further computations, the fact that / : H ⊗ A → H, . : H ⊗ A → A and f : H ⊗ H → A are coalgebra maps can be written explicitly as follows:

∆H (h / a) = h(1) / a(1) ⊗ h(2) / a(2), εA(h / a) = εH (h)εA(a) (2.1)

∆A(h . a) = h(1) . a(1) ⊗ h(2) . a(2), εA(h . a) = εH (h)εA(a) (2.2) ∆A f(g, h) = f(g(1), h(1)) ⊗ f(g(2), h(2)), εA f(g, h) = εH (g)εH (h) (2.3) for all g, h ∈ H, a ∈ A. A For a Hopf algebra A we denote by AM the category of left-left A-Hopf modules : the objects A are triples (M, ·, ρ), where (M, ·) ∈ AM is a left A-module, (M, ρ) ∈ M is a left A-comodule such that ρ(a · m) = a(1)m<−1> ⊗ a(2) · m<0> A co(A) for all a ∈ A and m ∈ M. For (M, ·, ρ) ∈ AM we denote by M = {m ∈ M | ρ(m) = 1A ⊗ m} the subspace of coinvariants . The fundamental theorem for Hopf modules states that for any A-Hopf module M the canonical map ϕ : A ⊗ M co(A) → M, ϕ(a ⊗ m) := a · m

55 CHAPTER 2. UNIFIED PRODUCTS for all a ∈ A and m ∈ M is bijective with the inverse given by

−1 co(A) −1 ϕ : M → A ⊗ M , ϕ (m) := m<−2> ⊗ S(m<−1>) · m<0> for all m ∈ M. A AYD will denote the pre-braided monoidal category of left-left Yetter-Drinfel’d modules over A A: the objects are triples (M, ·, ρ), where (M, ·) ∈ AM is a left A-module, (M, ρ) ∈ M is a left A-comodule such that

ρ(a · m) = a(1)m<−1>S(a(3)) ⊗ a(2) · m<0>

A for all a ∈ A and m ∈ M. If (L, ·, ρ) is a bialgebra in the monoidal category AYD then the Radford biproduct L ∗ A is the vector space L ⊗ A with the bialgebra structure given by

(l ∗ a)(m ∗ b) := l(a(1) · m) ∗ a(2) b

∆(l ∗ a) := l(1) ∗ l(2)<−1>a(1) ⊗ l(2)<0> ∗ a(2) for all l, m ∈ L and a, b ∈ A, where we denoted l ⊗ a ∈ L ⊗ A by l ∗ a. L ∗ A is a bialgebra with the unit 1L ∗ 1A and the counit εL∗A(l ∗ a) = εL(l)εA(a), for all l ∈ L and a ∈ A.

2.2 The extending structures problem

Let C be a category whose objects are sets endowed with various algebraic structures (S) and D be a category such that there exists a forgetful functor F : C → D, i.e. a functor that forgets some of the structures (S). To illustrate, the following are forgetful functors:

F : Gr → Set, F : Lie → Vec, F : Hopf → CoAlg, F : Hopf → Alg where Gr, Set, Lie, Vec, Hopf, CoAlg, Alg are the categories of all groups, sets, Lie algebras, vector spaces, Hopf algebras, coalgebras and respectively algebras. In this context we formulate a general problem which may be of interest for many areas of mathematics: Extending Structures Problem (ES): Let F : C → D be a forgetful functor and consider two objects C ∈ C, D ∈ D such that F (C) is a subobject of D in D. Describe and classify all mathematical structures (S) that can be defined on D such that D becomes an object of C and C is a subobject of D in the category C (the classification is up to an isomorphism that stabilizes C and a certain type of fixed quotient D/C). The ES-problem generalizes and unifies two famous and still open problems in the theory of groups: the extension problem of Holder¨ [73] and the factorization problem of Ore [115]. Let us explain this. In [16] we formulated the ES-problem at the level of groups, corresponding to the forgetful functor F : Gr → Set: if A is a group and E a set such that A ⊆ E [16, Corollary 2.10], describe all group structures (E, ·) that can be defined on the set E such that A is a subgroup of (E, ·). In order to do that we have introduced a new product for groups, called the unified product ([16, Theorem 2.6]), such that both the crossed product (the tool for

56 2.3. EXTENDING STRUCTURES: THE GROUP CASE the extension problem) and the bicrossed product (the tool for the factorization problem) of two groups are special cases of it. The unified product for groups is associated to a group A and a new hidden algebraic structure (H, ∗), connected by two actions and a generalized cocycle satisfying some compatibility conditions. We now take a step forward and formulate the ES-problem at the level of Hopf algebras corresponding to the forgetful functor F : Hopf → CoAlg: (H-C) Extending Structures Problem: Let A be a Hopf algebra and E a coalgebra such that A is a subcoalgebra of E. Describe and classify all Hopf algebra structures that can be defined on E such that A is a Hopf subalgebra of E. There is of course a dual version of the ES-problem corresponding to the forgetful functor F : Hopf → Alg to be addressed somewhere else. If at the level of groups the ES-problem is elementary, for Hopf algebras the problem is more difficult. Indeed, let A be a group and E a set such that A ⊆ E. For a field k we look at the extension k[A] ⊆ k[E], where k[A] is the group algebra that is a Hopf algebra and a subcoalgebra in the group-like coalgebra k[E]. Assume now that (E, ·) is a group structure on the set E such that A is a subgroup of (E, ·). Thus, we obtain an extension of Hopf algebras k[A] ⊆ k[E]. This extension of Hopf algebras has a remarkable property: let H ⊆ E be a system of representatives for the right cosets of the subgroup A in the group (E, ·) such that 1E ∈ H. Since the map u : A × H → E, u(a, h) = a · h is bijective, we obtain that the multiplication map

k[A] ⊗ k[H] → k[E], a ⊗ h 7→ a · h is bijective, i.e. the Hopf algebra k[E] factorizes through the Hopf subalgebra k[A] and the subcoalgebra k[H]. This is not valid for arbitrary extensions of Hopf algebras. Therefore, we have to restrict the (H-C) extending structures problem to those Hopf algebras E that factorize through a given Hopf subalgebra A and a given subcoalgebra H: we called this the restricted (H-C) ES-problem and we shall give a complete answer to it in the present chapter. It turns out that H is not only a subcoalgebra of E but will be endowed additionally with a hidden algebraic structure that will play the role of the system of representatives for congruence in the theory of groups.

2.3 Extending structures: the group case

At the level of groups, i.e. corresponding to the forgetful functor F : Gr → Set from the category of groups to the category of sets, the ES problem has a very tempting statement: (Gr) Extending structures problem. Let H be a group and E a set such that H ⊆ E. Describe and classify up to an isomorphism that stabilizes H the set of all group structures · that can be deﬁned on E such that H is a subgroup of (E, ·). In other words, the (Gr) ES-problem is trying to provide an answer to the very natural question: to what extent a group structure on H can be extended beyond H to a bigger set which contains H as a subset in such a way that H would become a subgroup within the new structure. As we showed above, this problem can be stated in many other ﬁelds of study such as: Lie groups or Lie

57 CHAPTER 2. UNIFIED PRODUCTS algebras, algebras or coalgebras, Hopf algebras and quantum groups, locally compact groups or locally compact quantum groups etc. The (Gr) ES-problem generalizes and unifies two famous problems in the theory of groups which served as models for our construction: the extension problem of Holder¨ [73] and the factorization problem of Ore [115]. Let us explain this briefly. Consider two groups H and G. The extension problem of Holder¨ consists of describing and classifying all groups E containing H as a normal subgroup such that E/H =∼ G. An important step related to the extension problem was made by Schreier: the crossed product associated to a crossed system (H, G, α, f) of groups was constructed and it was proven that any extension E of H by G is equivalent to a crossed product extension. The extension problem was one of the most studied problems in group theory in the last century and it has been the starting point of new subjects in mathematics such as cohomology of groups, homological algebra, crossed products of groups acting on algebras, crossed products of Hopf algebras acting on algebras, crossed products for von Neumann algebras etc. One of the most important contributions to the extension problem was given by S. Eilenberg and S. MacLane in two fundamental papers [67]. The factorization problem is a ”dual” of the extension problem and it was formulated by Ore [115] but its roots descend to E. Maillet’s paper [94]. It consists of describing and classifying up to an isomorphism all groups E that factorize through H and G: i.e. E contains H and G as subgroups such that E = HG and H ∩ G = {1}. The dual version of Schreier’s theorem was proven by Takeuchi [143]: the bicrossed product associated to a matched pair of groups (H, G, /, .) was constructed and it was proven that a group E factorizes through H and G if and only if E is isomorphic to a bicrossed product H ./ G. The factorization problem is even more difficult than the more popular extension problem and little progress has been made since then. For instance, in the case of two cyclic groups H and G, not both finite, the problem was started by L. Redei´ in [132] and finished by P.M. Cohn in [56]. If H and G are both finite cyclic groups the problem seems to be still open, even though J. Douglas [65] has devoted four papers to the subject. The case of two cyclic groups, one of them being of prime order, was solved recently in [11]. In the construction of a crossed product associated to a crossed system a weak action α : G → Aut (H) and an α-cocycle f : G × G → H are used, while the construction of a bicrossed product involves two compatible actions . : G × H → H and / : G × H → G. Even if their starting points are different, the two constructions have something in common: the crossed product structure as well as the bicrossed product structure are defined on the same set, namely H ×G. Moreover, H =∼ (H, 1) is a subgroup in both the crossed as well as the bicrossed product (in fact H =∼ (H, 1) is even a normal subgroup in the crossed product).

The reconstruction of a group from a subgroup.

The abstract deﬁnition of the uniﬁed product of groups will arise from the following elementary question subsequent to the (Gr) ES-problem: Let H ≤ E be a subgroup in E that is not necessary normal. Can we reconstruct the group structure on E from the one of H and some extra set of datum? The next theorem indicates the way we can perform this reconstruction. Moreover, it indicates

58 2.3. EXTENDING STRUCTURES: THE GROUP CASE a way to generalize our construction to other mathematical objects like: Hopf algebras, Lie algebras, Lie groups, compact quantum groups, etc.

Theorem 2.3.1 Let H ≤ E be a subgroup of a group E. Then:

1. There exists a map p : E → H such that p(1) = 1 and

p(h x) = h p(x) (2.4) for all h ∈ H and x ∈ E.

−1 2. For such a map p : E → H we deﬁne S = Sp := p (1) = {x ∈ E | p(x) = 1}. Then the multiplication map ϕ : H × S → E, ϕ(h, s) := hs (2.5) for all h ∈ H and s ∈ S is bijective with the inverse given by ϕ−1 : E → H × S, ϕ−1(x) = p(x), p(x)−1 x for all x ∈ E.

3. For p and S as above there exist four maps . = .p : S × H → H, / = /p : S × H → S, f = fp : S × S → H and ∗ = ∗p : S × S → S given by the formulas s . h := p(sh), s / h := p(sh)−1sh −1 f(s1, s2) := p(s1s2), s1 ∗ s2 := p(s1s2) s1s2

0 0 for all s, s1, s2 ∈ S and h ∈ H. Using these maps, the unique group structure · on the set H × S such that ϕ :(H × S, ·) → E is an isomorphism of groups is given by: (h1, s1) · (h2, s2) := h1(s1 . h2)f(s1 / h2, s2), (s1 / h2) ∗ s2 (2.6)

for all h1, h2 ∈ H and s1, s2 ∈ S.

Proof: (1) Using the axiom of choice we can fix Γ = (xi)i∈I ⊂ E to be a system of representatives for the right congruence modulo H in E such that 1 ∈ Γ. Then for any x ∈ E there exists an unique hx ∈ H and an unique xi0 ∈ Γ such that x = hx xi0 . Thus, there exists a well defined map p : E → H given by the formula p(x) := hx, for all x ∈ E. As 1 ∈ Γ we have that p(1) = 1. Moreover, for any h ∈ H and x ∈ E we have that hx = hhx xi0 . Thus p(hx) = hhx = hp(x), as needed. (2) We note that p(x)−1x ∈ S as pp(x)−1 x= p(x)−1p(x) = 1, for all x ∈ E. The rest is straightforward. (3) First we note that / and ∗ are well defined maps. Indeed, using (2.4), for any s ∈ S and h ∈ H we have: p(s / h) = pp(sh)−1sh= p(sh)−1p(xh) = 1, i.e. s / h ∈ S. In the same way p(s1 ∗ s2) = 1, for any s1, s2 ∈ S. Now, we shall prove the following two formulas:

p(s1h2s2) = (s1 . h2)f(s1 / h2, s2) (2.7)

59 CHAPTER 2. UNIFIED PRODUCTS and −1 (s1 / h2) ∗ s2 = p(s1h2s2) s1h2s2 (2.8) for all s1, s2 ∈ S and h2 ∈ H. Indeed, (s1 . h2)f(s1 / h2, s2) = p(s1h2) p (s1 / h2)s2 −1 = p(s1h2) p p(s1h2) s1h2s2 (2.4) = p(s1h2s2) and

−1 (s1 / h2) ∗ s2 = p(s1h2) s1h2 ∗s2 −1 −1 −1 = p p(s1h2) s1h2s2 p(s1h2) s1h2s2

(2.4) −1 −1 −1 = [p(s1h2) p(s1h2s2)] p(s1h2) s1h2s2 −1 = p(s1h2s2) s1h2s2 as needed. Now, ϕ : H × S → E is a bijection between the set H × S and the group E. Thus, there exists a unique group structure · on the set H ×S such that ϕ is an isomorphisms of groups. This group structure is obtained by transferring the group structure from E via the bijection ϕ, i.e. is given by:

−1 −1 (h1, s1) · (h2, s2) = ϕ ϕ(h1, s1)ϕ(h2, s2) = ϕ (h1s1h2s2) −1 = p(h1s1h2s2), p(h1s1h2s2) h1s1h2s2 (2.4) −1 = h1p(s1h2s2), p(s1h2s2) s1h2s2 (2.7),(2.8) = h1(s1 . h2)f(s1 / h2, s2), (s1 / h2) ∗ s2 for all h1, h2 ∈ H and s1, s2 ∈ S.

Remarks 2.3.2 1. Theorem 2.3.1 contains as a special case the fact that split monomorphism in the category of groups are described by semidirect products of groups. Indeed, let H ≤ E be a subgroup of E such that the inclusion i : H → E is a split monomorphism of groups, that is there exists p : E → H a morphism of groups such that p(h) = h, for all h ∈ H. Any such morphism p satisﬁes the condition (2.4) and hence S := Ker(p) is a normal subgroup of E. Moreover, the maps constructed in (3) of Theorem 2.3.1 are the following: . and f are the trivial maps, ∗ is exactly the subgroup structure of S in E and / is the action by conjugation, i.e. s / h = h−1sh. It follows from (2.6) that the multiplication on H × S is exactly the one of the semidirect product H o S, in the right convention, namely: (h1, s1) · (h2, s2) = h1h2, (s1 / h2)s2 for all h1, h2 ∈ H and s1, s2 ∈ S. Thus ϕ : H o S → E is an isomorphism of groups.

60 2.3. EXTENDING STRUCTURES: THE GROUP CASE

2. As 1 ∈ S and p(s) = 1, for all s ∈ S the maps . = .p, / = /p, f = fp and ∗ = ∗p constructed in (3) of Theorem 2.3.1 satisfy the following normalization conditions:

s . 1 = 1, 1 . h = h, 1 / h = 1, s / 1 = s (2.9)

f(s, 1) = f(1, s) = 1, s ∗ 1 = 1 ∗ s = s (2.10) for all s ∈ S and h ∈ H. Hence, the multiplication ∗ on S has a unit but is not necessary associative. In fact, we can easily prove that it satisﬁes the following compatibility: (s1 ∗ s2) ∗ s3 = s1 / f(s2, s3) ∗(s2 ∗ s3) (2.11) for all s1, s2, s3 ∈ S, i.e. ∗ is associative up to the pair (/, f). Moreover, any element s ∈ S is left invertible in (S, ∗); more precisely we can show that for any s ∈ S there exists a unique element s0 ∈ S such that s0 ∗ s = 1. Indeed, the multiplication given by (2.6) is a group structure on H × S having (1, 1) as a unit. In particular, for any s ∈ S there exists a unique h0 ∈ H and s0 ∈ S such that (h0, s0) · (1, s) = (1, 1). Thus, taking into account the normalizing conditions (2.9) we obtain that (h0f(s0, s), s0 ∗ s) = (1, 1), i.e. in particular it follows that s0 ∗ s = 1.

The abstract construction of the uniﬁed product of groups

Let H be a group and E a set such that H ⊆ E. Theorem 2.3.1 describes the way any group structure · on the set E such that H is a subgroup of (E, ·) should look like. It remains to show what type of abstract axioms should the system of maps (∗, /, ., f) satisfy such that (2.6) is indeed a group structure. This will be done bellow.

Deﬁnition 2.3.3 Let H be a group. An extending datum of H is a system Ω(H) = (S, 1S, ∗), /, ., f where:

(1) (S, 1S) is a pointed set, ∗ : S × S → S is a binary operation such that for any s ∈ S

s ∗ 1S = 1S ∗ s = s (2.12)

(2) The maps / : S × H → S, . : S × H → H and f : S × S → H satisfy the following normalization conditions for any s ∈ S and h ∈ H:

s / 1H = s, 1S / h = 1S, 1S . h = h, s . 1H = 1H , f(s, 1S) = f(1S, s) = 1H (2.13)

Let H be a group and Ω(H) = (S, 1S, ∗), /, ., f an extending datum of H. We denote by H nΩ(H) S := H n S the set H × S with the binary operation deﬁned by the formula: (h1, s1) · (h2, s2) := h1(s1 . h2)f(s1 / h2, s2), (s1 / h2) ∗ s2 (2.14) for all h1, h2 ∈ H and s1, s2 ∈ S.

61 CHAPTER 2. UNIFIED PRODUCTS

Deﬁnition 2.3.4 Let H be a group and Ω(H) = (S, 1S, ∗), /, ., f an extending datum of H. The object H n S introduced above is called the uniﬁed product of H and Ω(H) if H n S is a group with the multiplication given by (2.14). In this case the extending datum Ω(H) is called a group extending structure of H. The maps . and / are called the actions of Ω(H) and f is called the (., /)-cocycle of Ω(H).

The multiplication formula deﬁned by (2.14) arise naturally from (3) of Theorem 2.3.1 which was our starting point on solving the extending structures problem for groups.

Remark 2.3.5 Using the normalizing conditions (2.12) and (2.13) it is straightforward to prove that (1H , 1S) is a unit of the multiplication (2.14) and the following cross relations hold in H n S:

(h1, 1S) · (h2, s2) = (h1h2, s2) (2.15)

(h1, s1) · (1H , s2) = (h1f(s1, s2), s1 ∗ s2) (2.16)

(h1, s1) · (h2, 1S) = (h1(s1 . h2), s1 / h2) (2.17) for all h1, h2 ∈ H and s1, s2 ∈ S.

Next, we indicate the abstract system of axioms that need to be satisﬁed by the functions (∗, /, ., f) such that H n S becomes a uniﬁed product.

Theorem 2.3.6 Let H be a group and Ω(H) = (S, 1S, ∗), /, ., f an extending datum of H. The following statements are equivalent: (1) H n S is an uniﬁed product;

(2) The following compatibilities hold for any s, s1, s2, s3 ∈ S and h, h1, h2 ∈ H:

(ES1) The map / : S × H → S is a right action of the group H on the set S;

(ES2) (s1 ∗ s2) ∗ s3 = s1 / f(s2, s3) ∗(s2 ∗ s3)

(ES3) s . (h1h2) = (s . h1) (s / h1) . h2

(ES4) (s1 ∗ s2) / h = s1 / (s2 . h) ∗(s2 / h)

(ES5) s1 . (s2 . h) f s1 / (s2 . h), s2 / h = f(s1, s2) (s1 ∗ s2) . h

(ES6) f(s1, s2)f(s1 ∗ s2, s3) = s1 . f(s2, s3) f s1 / f(s2, s3), s2 ∗ s3

62 2.3. EXTENDING STRUCTURES: THE GROUP CASE

0 0 (ES7) For any s ∈ S there exists s ∈ S such that s ∗ s = 1S.

Proof: We know that (1H , 1S) is a unit for the operation (2.14). We prove now that the operation · given by (2.14) is associative if and only if the compatibility conditions (ES1) − (ES6) hold. Assume ﬁrst that · is associative and let h, h1, h2 ∈ H and s, s1, s2 ∈ S. The associativity condition

[(1H , s) · (h1, 1S)] · (h2, 1S) = (1H , s) · [(h1, 1S) · (h2, 1S)]

gives, after we use the cross relations (2.17) and (2.15), (s . h1, s / h1) · (h2, 1S) = (1H , s) · (h1h2, 1S). Thus (s . h1) (s / h1) . h2 , (s / h1) / h2 = s . (h1h2), s / (h1h2) and hence (ES1) and (ES3) hold. Now, we write the associativity condition [(1H , s1) · (1H , s2)] · (1H , s3) = (1H , s1) · [(1H , s2) · (1H , s3)] and compute this equality using the cross relation (2.16) we obtain the fact that the compatibility conditions (ES2) and (ES6) hold. Finally, if we write the associativity condition [(1H , s1) · (1H , s2)] · (h, 1S) = (1H , s1) · [(1H , s2) · (h, 1S)] and use (2.16) and then (2.17) we obtain precisely the fact that (ES4) and (ES5) hold.

Conversely, assume that the compatibility conditions (ES1) − (ES6) hold. Then for any h1, h2, h3 ∈ H and s1, s2, s3 ∈ S we have:

(h1, s1) · [(h2, s2) · (h3, s3)] = = h1[s1 . (h2(s2 . h3)f(s2 / h3, s3))]f s1 / (h2(s2 . h3)f(s2 / h3, s3)), (s2 / h3) ∗ s3 , [s1 / (h2(s2 . h3)f(s2 / h3, s3))] ∗ ((s2 / h3) ∗ s3)) (ES3) = h1(s1 . h2)[(s1 / h2) . ((s2 . h3)f(s2 / h3, s3))]f s1 / (h2(s2 . h3)f(s2 / h3, s3)), (s2 / h3) ∗ s3 , [s1 / (h2(s2 . h3)f(s2 / h3, s3))] ∗ ((s2 / h3) ∗ s3)) (ES1) = h1(s1 . h2)[(s1 / h2) . ((s2 . h3)f(s2 / h3, s3))]f (s1 / h2) / [(s2 . h3)f(s2 / h3, s3)], (s2 / h3) ∗ s3 , [s1 / (h2(s2 . h3)f(s2 / h3, s3))] ∗ ((s2 / h3) ∗ s3))

(ES3) = h1(s1 . h2)((s1 / h2) . (s2 . h3))[((s1 / h2) / (s2 . h3)) . f(s2 / h3, s3)] f (s1 / h2) / [(s2 . h3)f(s2 / h3, s3)], (s2 / h3) ∗ s3 , [s1 / (h2(s2 . h3)f(s2 / h3, s3))] ∗ ((s2 / h3) ∗ s3))

(ES5) −1 = h1(s1 . h2)f(s1 / h2, s2)[((s1 / h2) ∗ s2) . h3]f((s1 / h2) / (s2 . h3), s2 / h3) [((s1 / h2) / (s2 . h3)) . f(s2 / h3, s3)]f (s1 / h2) / [(s2 . h3)f(s2 / h3, s3)], (s2 / h3) ∗ s3 , [s1 / (h2(s2 . h3)f(s2 / h3, s3))] ∗ ((s2 / h3) ∗ s3))

63 CHAPTER 2. UNIFIED PRODUCTS

(ES6) −1 = h1(s1 . h2)f(s1 / h2, s2)[((s1 / h2) ∗ s2) . h3]f((s1 / h2) / (s2 . h3), s2 / h3) f((s1 / h2) / (s2 . h3), s2 / h3)f [(s1 / h2) / (s2 . h3)] ∗ (s2 / h3), s3 , [s1 / (h2(s2 . h3)f(s2 / h3, s3))] ∗ ((s2 / h3) ∗ s3)) = h1(s1 . h2)f(s1 / h2, s2)[((s1 / h2) ∗ s2) . h3]f [(s1 / h2) / (s2 . h3)] ∗ (s2 / h3), s3 , [s1 / (h2(s2 . h3)f(s2 / h3, s3))] ∗ ((s2 / h3) ∗ s3)) (ES4) = h1(s1 . h2)f(s1 / h2, s2)[((s1 / h2) ∗ s2) . h3]f [(s1 / h2) ∗ s2] / h3, s3 , [s1 / (h2(s2 . h3)f(s2 / h3, s3))] ∗ ((s2 / h3) ∗ s3)) (ES1) = h1(s1 . h2)f(s1 / h2, s2)[((s1 / h2) ∗ s2) . h3]f [(s1 / h2) ∗ s2] / h3, s3 , [s1 / (h2(s2 . h3))] / f(s2 / h3, s3)] ∗ [(s2 / h3)] ∗ s3] (ES2) = h1(s1 . h2)f(s1 / h2, s2)[((s1 / h2) ∗ s2) . h3]f [(s1 / h2) ∗ s2] / h3, s3 , [((s1 / (h2(s2 . h3))) ∗ (s2 / h3)] ∗ s3 (ES1) = h1(s1 . h2)f(s1 / h2, s2)[((s1 / h2) ∗ s2) . h3]f [(s1 / h2) ∗ s2] / h3, s3 , [((s1 / h2) / (s2 . h3)) ∗ (s2 / h3)] ∗ s3 (ES4) = h1(s1 . h2)f(s1 / h2, s2)[((s1 / h2) ∗ s2) . h3]f [(s1 / h2) ∗ s2] / h3, s3 , [((s1 / h2) ∗ s2) / h3] ∗ s3 = [(h1, s1) · (h2, s2)] · (h3, s3) as needed. To conclude, we proved that (H n S, ·) is a monoid if and only if (ES1) − (ES6) hold. Assume now that (H n S, ·) is a monoid: it remains to be proved that the monoid is actually a group if and only if (ES7) holds. Indeed, in the monoid (H n S, ·) we have:

(h, 1S) · (1H , s) = (h, s), (h1, 1S) · (h2, 1S) = (h1h2, 1S) for all h, h1, h2 ∈ H and s ∈ S. In particular, any element of the form (h, 1S), for h ∈ H is invertible in (H n S, ·). Now, a monoid is a group if and only if each of his elements has a left inverse. As · is associative it follows from:

−1 (h , 1S) · (h, s) = (1H , s) that (H n S, ·) is a group if and only if (1H , s) has a left inverse for all s ∈ S. Hence, for any s ∈ S there exist elements s0 ∈ S and h0 ∈ H such that

0 0 0 0 0 (h , s ) · (1H , s) = h f(s , s), s ∗ s = (1H , 1S)

0 0 0 −1 This is of course equivalent to the fact that s ∗ s = 1S for all s ∈ S and h = f s , s .

64 2.3. EXTENDING STRUCTURES: THE GROUP CASE

The proof is now ﬁnished. We note that the inverse of an element (h, s) in the group (H n S, ·) is given by the formula

−1 0 −1 0 −1 0 −1 0 −1 0 −1 (h, s) = f(s , s) , s · (h , 1S) = f(s , s) (s . h ), s / h

0 where s ∗ s = 1S.

Next we provide two universal properties satisfied by the unified product. A different approach, from category theory point of view, is presented in [16, Theorem 2.15] where the unified product is proved to be an initial object in a certain category Ω(H)C and also a final object in another category DΩ(H) (which is not the dual of Ω(H)C).

Proposition 2.3.7 Let H be a group and Ω(H) = (S, 1S, ∗), /, ., f a group extending structure of H. Then:

(1) For any group G, any maps u : G → H, v : G → S such that for all x, y ∈ G we have:

u(xy) = u(x)[v(x) . u(y)]f(v(x) / u(y), v(y)) (2.18) v(xy) = [v(x) / u(y)] ∗ v(y) (2.19)

there exists a unique morphism of groups ψ : G → H nΩ S such that the following diagram commutes: πH πS H o H nΩ S / S (2.20) dHH O v; H vv HH ψ v u HH vvv HH vv H vv G

(2) For any group G, any morphism of groups u : H → G and any map v : S → G such that for all s, s1, s2 ∈ S and h ∈ H we have:

v(s1)v(s2) = u(f(s1, s2))v(s1 ∗ s2) (2.21) v(s)u(h) = u(s . h)v(s / h) (2.22)

there exists a unique morphism of groups φ : H nΩ S → G such that the following diagram commutes:

iH iS H / H nΩ S o S (2.23) H v HH vv HH φ v u HH vvv HH vv H vv $ G {

Proof: (1) Let G be a group and u : G → H, v : G → S to maps satisfying (2.18) and (2.19). We have to prove that there exists a unique morphism of groups ψ : G → H nΩ S such that (2.20) commutes. Assume ﬁrst that ψ satisﬁes the condition above. From the commutativity of the diagram we obtain: πG ◦ψ = u and πS ◦ψ = v, i.e. ψ(g) = (u(g), v(g)), and we proved that

65 CHAPTER 2. UNIFIED PRODUCTS

ψ is uniquely determined by u and v. The existence of ψ can be proved as follows: we deﬁne ψ : G → H nΩ S by ψ(g) := (u(g), v(g)). It follows from here that:

ψ(g1) · ψ(g2) = (u(g1), v(g1)) · (u(g2), v(g2))

= (u(g1)(v(g1) . u(g2))f(v(g1) / u(g2), v(g2)), (v(g1) / u(g2)) ∗ v(g2)) (2.18),(2.19) = (u(g1g2), v(g1g2))

= ψ(g1g2) thus ψ is a morphism of groups. The commutativity of the diagram is obvious.

(2) Let G be a group, u : H → G a morphism of groups and v : S → G a map satisfying (2.21) and (2.22). We need to prove that there exists an unique morphism of groups φ : H nΩ S → G such (2.23) commutes. Assume first that φ satisfies the above condition. We obtain: φ((h, s)) = φ((h, 1S) · (1H , s)) = φ((h, 1S))φ((1H , s)) = (φ ◦ iH )(h)(φ ◦ iS)(s)) = u(h)v(s), for all h ∈ H, s ∈ S and we proved that φ is uniquely determined by u and v. The existence of φ can be proved as follows: we define φ : H nΩ S → G by φ((h, s)) := u(h)v(s), for all h ∈ H and s ∈ S. It follows that:

φ((h1, s1) · (h2, s2)) = φ(h1(s1 . h2)f(s1 / h2, s2), (s1 / h2) ∗ s2)

= u(h1)u(s1 . h2)u(f(s1 / h2, s2))v((s1 / h2) ∗ s2) (2.21) = u(h1)u(s1 . h2)v(s1 / h2)v(s2) (2.22) = u(h1)v(s1)u(h2)v(s2)

= φ((h1, s1))φ((h2, s2)) thus φ is a morphism of groups. The commutativity of the diagram is obvious.

66 2.4. EXTENDING STRUCTURES: THE QUANTUM VERSION

2.4 Extending structures: the quantum version

To start with, we recall the classical constructions of the crossed product of a Hopf algebra H acting on an algebra A and of the bicrossed product (double cross product in Majid’s terminology) of two Hopf algebras H and A, as the product that we define will generalize both of them. In Section 2.5 we define the concept of an extending structure of a bialgebra A consisting of a system Ω(A) = H, /, ., f, where H is a coalgebra and an unitary not necessarily associative algebra such that A and H are connected by three coalgebra maps / : H ⊗ A → H, . : H ⊗ A → A, f : H ⊗ H → A satisfying some natural normalization conditions (Defini- tion 2.5.1). For a bialgebra extending structure Ω(A) = H, /, ., f of A we define a product AnΩ(A) H = AnH and call it the unified product: both the crossed product of an Hopf algebra acting on an algebra and the bicrossed product (double cross product in Majid’s terminology) of two Hopf algebras are special cases of the unified product. Theorem 2.5.4 gives necessary and sufficient conditions for A n H to be a bialgebra, which is precisely the Hopf algebra version of [16, Theorem 2.6] proven for the group case that served as a model for us. The seven compatibility conditions in Theorem 2.5.4 are very natural and, mutatis-mutandis, are the ones (with two reasonable deformations via the right action /) that appear in the construction of the crossed product and the bicrossed product of two Hopf algebras. Theorem 2.5.8 proves that a Hopf algebra E factorizes through a Hopf subalgebra A and a subcoalgebra H if and only if E is isomorphic to a unified product of A and H and gives the answer for the first part of the restricted (H-C) ES-problem. Section 2.6 is devoted to the classification part of the restricted (H-C) ES-problem. Our view point descends from the classical classification theorem of Schreier at the level of groups: all extensions of an abelian group K by a group Q are classified by the second cohomology group H2(Q, K) [133, Theorem 7.34]. Let A be a Hopf algebra. Two Hopf algebra extending structures Ω(A) = H, /, ., f and Ω0(A) = H,/0,.0, f 0 are called equivalent if there exists 0 ϕ : A n H → A n H a left A-module, a right H-comodule and a Hopf algebra map. As in 0 group extension theory we shall prove that any such morphism ϕ : A n H → A n H is an isomorphism and the following diagram

iA πH A / A ./ H / H

IdA ϕ IdH

iA πH A / A ./0 H / H

0 is commutative. Theorem 2.6.4 shows that any such morphism ϕ : An H → AnH is uniquely determined by a coalgebra lazy 1-cocycle: i.e. a unitary coalgebra map u : H → A such that:

h(1) ⊗ u(h(2)) = h(2) ⊗ u(h(1)) for all h ∈ H. Corollary 2.6.6 is the Schreier type classification theorem for unified products: the part of the second cohomology group from the theory of groups will be played now by a 2 special quotient set Hl,c(H, A, /). Also, a classification result for bicrossed product of two Hopf algebras is derived from Theorem 2.6.4.

67 CHAPTER 2. UNIFIED PRODUCTS

We recall here, for a further use, two classical constructions: the crossed product of a Hopf algebra acting on a k-algebra and the bicrossed product (double cross product in Majid’s terminology) of two Hopf algebras.

The crossed product of a Hopf algebra acting on an algebra

The crossed product of a Hopf algebra H acting on a k-algebra A was introduced independently in [35] and [62] as a generalization of the crossed product of groups acting on k-algebras. Let H be a Hopf algebra, A a k-algebra and two k-linear maps . : H ⊗ A → A, f : H ⊗ H → A such that h . 1A = εH (h)1A, 1H . a = a (2.24)

h . (ab) = (h(1) . a)(h(2) . b), f(h, 1H ) = f(1H , h) = εH (h)1A (2.25) for all h ∈ H, a, b ∈ A. The crossed product A#f H of A with H is the k-module A ⊗ H with the multiplication given by (a#h)(c#g) := a(h(1) . c)f h(2), g(1) # h(3)g(2) (2.26) for all a, c ∈ A, h, g ∈ H, where we denoted a⊗h by a#h. It can be proved [106, Lemma 7.1.2] that A#f H is an associative algebra with identity element 1A#1H if and only if the following two compatibility conditions hold: [g(1) . (h(1) . a)]f g(2), h(2) = f(g(1), h(1)) (g(2)h(2)) . a (2.27) g(1) . f(h(1), l(1)) f g(2), h(2)l(2) = f(g(1), h(1))f(g(2)h(2), l) (2.28) for all a ∈ A, g, h, l ∈ H. The ﬁrst compatibility is called the twisted module condition while (2.28) is called the cocycle condition. The crossed product A#f H was studied only as an algebra extension of A, being an essential tool in Hopf-Galois extensions theory.

The bicrossed product of Hopf algebras

The bicrossed product of Hopf algebras was introduced by Majid in [100, Proposition 3.12] under the name of double cross product. We shall adopt the name of bicrossed product from [81, Theorem 2.3]. A matched pair of bialgebras is a system (A, H, /, .), where A and H are bialgebras, / : H ⊗ A → H, . : H ⊗ A → A are coalgebra maps such that (A, .) is a left H-module coalgebra, (H,/) is a right A-module coalgebra and the following compatibility conditions hold:

1H / a = εA(a)1H , h . 1A = εH (h)1A (2.29) g . (ab) = (g(1) . a(1)) (g(2) / a(2)) . b (2.30) (gh) / a = g / (h(1) . a(1)) (h(2) / a(2)) (2.31)

g(1) / a(1) ⊗ g(2) . a(2) = g(2) / a(2) ⊗ g(1) . a(1) (2.32)

68 2.5. BIALGEBRA EXTENDING STRUCTURES AND UNIFIED PRODUCTS for all a, b ∈ A, g, h ∈ H. Let (A, H, /, .) be a matched pair of bialgebras; the bicrossed product A ./ H of A with H is the k-module A ⊗ H with the multiplication given by

(a ./ h) · (c ./ g) := a(h(1) . c(1)) ./ (h(2) / c(2))g (2.33) for all a, c ∈ A, h, g ∈ H, where we denoted a ⊗ h by a ./ h. A ./ H is a bialgebra with the coalgebra structure given by the tensor product of coalgebras and moreover, if A and H are Hopf algebras, then A ./ H has an antipode given by the formula:

S(a ./ h) := (1A ./ SH (h)) · (SA(a) ./ 1H ) (2.34) for all a ∈ A and h ∈ H [101, Theorem 7.2.2].

2.5 Bialgebra extending structures and uniﬁed products

In this section we shall introduce the uniﬁed product for bialgebras; this will be the tool for answering the restricted (H-C) ES-problem. First we need the following:

Deﬁnition 2.5.1 Let A be a bialgebra. An extending datum of A is a system Ω(A) = H, /, ., f where: (i) H = H, ∆H , εH , 1H , · is a k-module such that H, ∆H , εH is a coalgebra, H, 1H , · is an unitary, not necessarily associative k-algebra, such that

∆H (1H ) = 1H ⊗ 1H (2.35)

(ii) The k-linear maps / : H ⊗ A → H, . : H ⊗ A → A, f : H ⊗ H → A are morphisms of coalgebras such that the following normalization conditions hold:

h . 1A = εH (h)1A, 1H . a = a, 1H / a = εA(a)1H , h / 1A = h (2.36)

f(h, 1H ) = f(1H , h) = εH (h)1A (2.37) for all h ∈ H, a ∈ A.

Let A be a bialgebra and Ω(A) = H, /, ., f an extending datum of A. We denote by AnΩ(A) H = A n H the k-module A ⊗ H together with the multiplication: (a n h) • (c n g) := a(h(1) . c(1))f h(2) / c(2), g(1) n (h(3) / c(3)) · g(2) (2.38) for all a, c ∈ A and h, g ∈ H, where we denoted a ⊗ h ∈ A ⊗ H by a n h.

Deﬁnition 2.5.2 Let A be a bialgebra and Ω(A) = H, /, ., f be an extending datum of A. The object A n H introduced above is called the uniﬁed product of A and Ω(A) if A n H is a

69 CHAPTER 2. UNIFIED PRODUCTS bialgebra with the multiplication given by (2.38), the unit 1A n 1H and the coalgebra structure given by the tensor product of coalgebras, i.e.:

∆AnH (a n h) = a(1) n h(1) ⊗ a(2) n h(2) (2.39)

εAnH (a n h) = εA(a)εH (h) (2.40) for all h ∈ H, a ∈ A. In this case the extending datum Ω(A) = (H, /, ., f) is called a bialgebra extending structure of A. The maps . and / are called the actions of Ω(A) and f is called the (., /)-cocycle of Ω(A). A bialgebra extending structure Ω(A) = (H, /, ., f) is called a Hopf algebra extending structure of A if A n H has an antipode.

The multiplication given by (2.38) has a rather complicated formula; however, for some speciﬁc elements we obtain easier forms which will be useful for future computations.

Lemma 2.5.3 Let A be a bialgebra and Ω(A) = H, /, ., f an extending datum of A. The following cross-relations hold:

(a n 1H ) • (c n g) = ac n g (2.41) (a n g) • (1A n h) = af(g(1), h(1)) n g(2) · h(2) (2.42) (a n g) • (b n 1H ) = a(g(1) . b(1)) n g(2) / b(2) (2.43)

Proof: Straightforward using the normalization conditions (2.35)-(2.37).

It follows from (2.41) that the map iA : A → A n H, iA(a) := a n 1H , for all a ∈ A, is a k-algebra map and (a n 1H ) • (1A n g) = a n g (2.44) for all a ∈ A and g ∈ H. Hence the set T := {a n 1H | a ∈ A} ∪ {1A n g | g ∈ H} is a system of generators as an algebra for A n H and this observation will turn out to be essential in proving the next theorem which provides necessary and sufﬁcient conditions for A n H to be a bialgebra: it is the Hopf algebra version of [16, Theorem 2.6] where the uniﬁed product for groups is constructed.

Theorem 2.5.4 Let A be a bialgebra and Ω(A) = H, /, ., f an extending datum of A. The following statements are equivalent: (1) A n H is an uniﬁed product; (2) The following compatibilities hold:

(2a) ∆H : H → H ⊗ H and εH : H → k are k-algebra maps;

(2b) (H,/) is a right A-module structure;

70 2.5. BIALGEBRA EXTENDING STRUCTURES AND UNIFIED PRODUCTS

(BE1) (g · h) · l = g / f(h(1), l(1)) ·(h(2) · l(2))

(BE2) g . (ab) = (g(1) . a(1))[(g(2) / a(2)) . b]

(BE3) (g · h) / a = [g / (h(1) . a(1))] · (h(2) / a(2))

(BE4) [g(1) . (h(1) . a(1))]f g(2) / (h(2) . a(2)), h(3) / a(3) = f(g(1), h(1))[(g(2) · h(2)) . a]

BE5) g(1) . f(h(1), l(1)) f g(2) / f(h(2), l(2)), h(3) · l(3) = f(g(1), h(1))f(g(2) · h(2), l)

(BE6) g(1) / a(1) ⊗ g(2) . a(2) = g(2) / a(2) ⊗ g(1) . a(1)

(BE7) g(1) · h(1) ⊗ f(g(2), h(2)) = g(2) · h(2) ⊗ f(g(1), h(1))

for all g, h, l ∈ H and a, b ∈ A.

Before going into the proof of the theorem, we have a few observations on the relations (BE1)− (BE7) in Theorem 2.5.4. Although they look rather complicated at first sight, they are in fact quite natural and can be interpreted as follows: (2a) and (2b) show that (H, ∆H , εH , 1H , ·) is a non-associative bialgebra and a right A-module coalgebra via /. (BE1) measures how far (H, 1H , ·) is from being an associative algebra. (BE2), (BE3) and (BE6) are exactly, mutatis- mutandis, the compatibility conditions (2.29)-(2.32) appearing in the definition of a matched pair of bialgebras. (BE4) and (BE5) are deformations via the action / of the twisted module condition (5) and respectively of the cocycle condition (2.28) which appears in the definition of the crossed product for Hopf algebras. (BE7) is a symmetry condition for the cocycle f similar to (BE6). Both relations are trivially fulfilled if, for example, H is cocommutative or f is the trivial cocycle.

Proof: We prove Theorem 2.5.4 in several steps. From (2.41) and (2.43) it is straightforward

that 1A n 1H is a unit for the algebra (A n H, •). Next, we prove that εAnH given by (2.40) is an algebra map if and only if εH : H → k is an algebra map. For h, g ∈ H we have:

(2.42) εAnH (1A n h) • (1A n g) = εAnH f(h(1), g(1)) n g(2) · h(2) (2.3) = εH (h(1))εH (g(1))εH (g(2) · h(2))

= εH (g · h)

and ε(1A nh)ε(1A ng) = εH (h)εH (g). Thus, if εAnH is a k-algebra map then εH is a k-algebra

71 CHAPTER 2. UNIFIED PRODUCTS

map. Conversely, suppose that εH is a k-algebra map. Then, we have: εAnH (a n h) • (c n g) = εA(a)εH (h(1))εA(c(1))εH (h(2))εA(c(2))εH (g(1))εH (h(3)) εA(c(3))εH (g(2))

= εA(a)εH (h)εA(c)εH (g)

= εAnH (a n h)εAnH (c n g) for all a, c ∈ A and h ∈ H i.e. εAnH is an algebra map.

The next step is to prove that ∆AnH is a k-algebra map if and only if ∆H : H → H ⊗ H is (2.35) a k-algebra map and the relations (BE6), (BE7) hold. Observe that ∆AnH (1A n 1H ) = 1A n 1H ⊗ 1A n 1H . Since T = {a n 1H | a ∈ A} ∪ {1A n g | g ∈ H} generates A n H as an algebra, ∆AnH is a k-algebra map if and only if ∆AnH (xy) = ∆AnH (x)∆AnH (y) for all x, y ∈ T . First, observe that: ∆AnH (a n 1H ) • (b n 1H ) = ∆AnH (ab n 1H ) = a(1)b(1) n 1H ⊗ a(2)b(2) n 1H = a(1) n 1H ⊗ a(2) n 1H b(1) n 1H ⊗ b(2) n 1H

= ∆AnH (a n 1H )∆AnH (b n 1H ) and ∆AnH (a n 1H ) • (1A n g) = ∆AnH (a n g) = a(1) n g(1) ⊗ a(2) n g(2) = a(1) n 1H ⊗ a(2) n 1H 1A n g(1) ⊗ 1A n g(2)

= ∆AnH (a n 1H )∆AnH (1A n g) for all a, b ∈ A, g ∈ H. There are two more relations to consider; for g, h ∈ H we have: (2.42) ∆AnH (1A n g) • (1A n h) = ∆AnH f(g(1), h(1)) n g(2) · h(2) (2.3) = f(g(1)(1), h(1)(1)) n (g(2) · h(2))(1) ⊗ f(g(1)(2), h(1)(2)) n (g(2) · h(2))(2) = f(g(1), h(1)) n (g(3) · h(3))(1) ⊗ f(g(2), h(2)) n (g(3) · h(3))(2) and ∆AnH (1A n g)∆AnH (1A n h) = 1A n g(1) ⊗ 1A n g(2) 1A n h(1) ⊗ 1A n h(2) (2.42) = f(g(1), h(1)) n g(2) · h(2) ⊗ f(g(3), h(3)) n g(4) · h(4) Thus ∆AnH (1A n g) • (1A n h) = ∆AnH (1A n g)∆AnH (1A n h) if and only if

f(g(1), h(1)) n (g(3) · h(3))(1) ⊗ f(g(2), h(2)) n (g(3) · h(3))(2) = = f(g(1), h(1)) n g(2) · h(2) ⊗ f(g(3), h(3)) n g(4) · h(4)

72 2.5. BIALGEBRA EXTENDING STRUCTURES AND UNIFIED PRODUCTS

We show now that this relation holds if and only if ∆H : H → H ⊗ H is a k-algebra map and (BE7) holds. Indeed, suppose ﬁrst that the above relation holds. By applying εA ⊗Id⊗εA ⊗Id to it we obtain ∆H (g · h) = g(1) · h(1) ⊗ g(2) · h(2), i.e. ∆H is a k-algebra map. Furthermore, if we apply εA ⊗Id⊗Id⊗εH to it we obtain g(1) ·h(1) ⊗f(g(2), h(2)) = g(2) ·h(2) ⊗f(g(1), h(1)), i.e. (BE7). Conversely, suppose that ∆H is a k-algebra map and (BE7) holds. We then have:

f(g(1), h(1)) n (g(3) · h(3))(1) ⊗ f(g(2), h(2)) n (g(3) · h(3))(2) = = f(g(1), h(1)) n g(3) · h(3) ⊗ f(g(2), h(2)) n g(4) · h(4) = f(g(1), h(1)) n g(2)(2) · h(2)(2) ⊗ f(g(2)(1), h(2)(1)) n g(3) · h(3) (BE7) = f(g(1), h(1)) n g(2)(1) · h(2)(1) ⊗ f(g(2)(2), h(2)(2)) n g(3) · h(3) = f(g(1), h(1)) n g(2) · h(2) ⊗ f(g(3), h(3)) n g(4) · h(4) as needed. To end with, for the last family of generators we have:

(2.43) ∆AnH (1A n g) • (a n 1H ) = ∆AnH (g(1) . a(1) n g(2) / a(2)) (2.1),(2.2) = g(1) . a(1) n g(3) / a(3) ⊗ g(2) . a(2) n g(4) / a(4) and ∆AnH (1A n g)∆AnH (a n 1H ) = 1A n g(1) ⊗ 1A n g(2) a(1) n 1H ⊗ a(2) n 1H (2.43) = g(1) . a(1) n g(2) / a(2) ⊗ g(3) . a(3) n g(4) / a(4) Thus ∆AnH (1A n g) • (a n 1H ) = ∆AnH (1A n g)∆AnH (a n 1H ) if and only if

g(1) . a(1) n g(3) / a(3) ⊗ g(2) . a(2) n g(4) / a(4) = = g(1) . a(1) n g(2) / a(2) ⊗ g(3) . a(3) n g(4) / a(4)

This relation is equivalent to the compatibility condition (BE6): indeed, by applying εA ⊗ Id ⊗ Id ⊗ εH to it we obtain (BE6). Conversely suppose that (2h) holds. Then:

g(1) . a(1) n g(3) / a(3)⊗g(2) . a(2) n g(4) / a(4) = (BE6) = g(1) . a(1) n g(2) / a(2) ⊗ g(3) . a(3) n g(4) / a(4) as needed.

To resume, we proved until now that ∆AnH and εAnH are k-algebra maps if and only if the relations (2a), (BE6), (BE7) hold. In what follows we shall prove, in the hypothesis that

∆AnH and εAnH are k-algebra maps, that the multiplication given by (2.38) is associative if and only if the compatibility conditions (2b) and (BE1) − (BE5) hold. This will end the proof. We make use again of the fact that T generates A n H as an algebra. Thus • is associative if and only if x • (y • z) = (x • y) • z, for all x, y, z ∈ T . To start with, we will prove that:

(a n 1H ) • (y • z) = [(a n 1H ) • y] • z

73 CHAPTER 2. UNIFIED PRODUCTS for all a ∈ A and y, z ∈ T . Indeed, we have:

(2.43) (a n 1H ) • (1A n g) • (b n 1H ) = (a n 1H ) • (g(1) . b(1) n g(2) / b(2)) (2.41) = a(g(1) . b(1)) n (g(2) / b(2))) (2.43) = (a n g) • (b n 1H ) = (a n 1H ) • (1A n g) •(b n 1H ) and

(2.42) (a n 1H ) • (1A n g) • (1A n h) = (a n 1H ) • (f(g(1), h(1)) n g(2) · h(2)) (2.41) = af(g(1), h(1)) n g(2) · h(2) (2.42) = (a n g) • (1A n h) (2.41) = (a n 1H ) • (1A n g) •(1A n h)

The other two possibilities for choosing the elements of T can also be proven by a straightforward computation. Thus • is associative if and only if (1A n g) • (y • z) = [(1A n g) • y] • z, for all g ∈ H, y, z ∈ T . First we note that: (1A n g) • (a n 1H ) • (1A n h) = (1A n g) • (a n h)

= (g(1) . a(1))f(g(2) / a(2), h(1)) n (g(3) / a(3)) · h(2) (2.42) = (g(1) . a(1) n g(2) / a(2)) • (1A n h) (2.43) = (1A n g) • (a n 1H ) •(1A n h)

On the other hand: (1A n g) • (b n 1H ) • (c n 1H ) = (1A n g) • (bc n 1H ) (2.43) = g(1) . (b(1)c(1)) n g(2) / (b(2)c(2)) and

(2.43) (1A n g) • (b n 1H ) •(c n 1H ) = (g(1) . b(1) n g(2) / b(2)) • (c n 1H ) (2.43) = (g(1) . b(1)) (g(2) / b(2)) . c(1) n(g(3) / b(3)) / c(2) Hence (1A n g) • (b n 1H ) • (c n 1H ) = (1A n g) • (b n 1H ) •(c n 1H ) if and only if

g(1) . (b(1)c(1)) n g(2) / (b(2)c(2)) = (g(1) . b(1)) (g(2) / b(2)) . c(1) n(g(3) / b(3)) / c(2) (2.45)

74 2.5. BIALGEBRA EXTENDING STRUCTURES AND UNIFIED PRODUCTS for all b, c ∈ A and g ∈ H. We show now that the relation (2.45) is equivalent to the compatibility conditions (2b) and (BE2). Indeed, by applying εA ⊗ Id and respectively Id ⊗ εH in (2.45) we obtain relations (2b) respectively (BE2). Conversely, suppose that relations (2b) and (BE2) hold. We then have: (g(1) . b(1)) (g(2) / b(2)) . c(1) n(g(3) / b(3)) / c(2) = = (g(1)(1) . b(1)(1)) (g(1)(2) / b(1)(2)) . c(1) n(g(2) / b(2)) / c(2) (BE2) = g(1) . (b(1)c(1)) ⊗ (g(2) / b(2)) / c(2) (2b) = g(1) . (b(1)c(1)) ⊗ g(2) / (b(2)c(2)) i.e. (2.45) holds. Now we deal with the last two cases. Since . is a coalgebra map we obtain:

(2.43) (1A n g) • (1A n h) • (a n 1H ) = (1A n g) • (h(1) . a(1) n h(2) / a(2)) (2.2) = g(1) . (h(1) . a(1)) f(g(2) / (h(2) . a(2)), h(4) / a(4))

n[g(3) / (h(3) . a(3))] · (h(5) / a(5)) and

(2.42) (1A n g) • (1A n h) •(a n 1H ) = f(g(1), h(1)) n g(2) · h(2) •(a n 1H ) (2.43) = f(g(1), h(1))[(g(2) · h(2)) . a(1)] n (g(3) · h(3)) / a(2) Thus (1A n g) • (1A n h) • (a n 1H ) = (1A n g) • (1A n h) •(a n 1H ) if and only if

g(1) . (h(1) . a(1)) f g(2) / (h(2) . a(2)), h(4) / a(4) n[g(3) / (h(3) . a(3))] · (h(5) / a(5))

= f(g(1), h(1))[(g(2) · h(2)) . a(1)] n (g(3) · h(3)) / a(2)

We shall prove, using (BE6), that this relation is equivalent to the compatibility conditions (BE3) and (BE4). Indeed, by applying Id ⊗ εH and respectively εA ⊗ Id to it we obtain (BE3) and respectively (BE4). Conversely, suppose that relations (BE3) respectively (BE4) hold. We denote LHS the left hand side of the above relation. We have: LHS = g(1) . (h(1) . a(1)) f g(2) / (h(2) . a(2)), h(3)(2) / a(3)(2) n [g(3) / (h(3)(1) . a(3)(1))] · (h(4) / a(4)) (BE6) = g(1) . (h(1) . a(1)) f(g(2) / (h(2) . a(2)), h(3) / a(3)) n

[g(3) / (h(4) . a(4))] · (h(5) / a(5)) (BE4) = f(g(1), h(1))[(g(2) · h(2)) . a(1)] ⊗ [g(3) / (h(3) . a(2))] · (h(4) / a(3)) (BE3) = f(g(1), h(1))[(g(2) · h(2)) . a(1)] ⊗ (g(3) · h(3)) / a(2)

75 CHAPTER 2. UNIFIED PRODUCTS as needed. Only one associativity relation remains to be veriﬁed:

(2.42) (1A n g) • (1A n h) • (1A n l) = (1A n g) • (f(h(1), l(1)) n h(2) · l(2)) (2.3) = g(1) . f(h(1), l(1)) f g(2) / f(h(2), l(2)), h(4) · l(4) n g(3) / f(h(3), l(3)) ·(h(5) · l(5)) and (2.42) (1A n g) • (1A n h) •(1A n l) = f(g(1), h(1)) n g(2) · h(2) •(1A n l) (2.42) = f(g(1), h(1))f g(2) · h(2), l(1) n(g(3) · h(3)) · l(2) Hence (1A n g) • (1A n h) • (1A n l) = (1A n g) • (1A n h) •(1A n l) if and only if g(1) . f(h(1), l(1)) f g(2) / f(h(2), l(2)), h(4) · l(4) ⊗ g(3) / f(h(3), l(3)) ·(h(5) · l(5)) = = f(g(1), h(1))f g(2) · h(2), l(1) ⊗(g(3) · h(3)) · l(2) for all g, h, l ∈ H. We shall prove, using (BE7), that this relation is equivalent to the compatibility conditions (BE1) and (BE5). Indeed, by applying Id ⊗ εH and respectively εA ⊗ Id to it we obtain (BE1) and respectively (BE5). Conversely, suppose that relations (BE1) and (BE5) hold and denote LHS0 the left hand side of the above relation. Then:

0 LHS = g(1) . f(h(1), l(1)) f g(2) / f(h(2), l(2)), h(3)(2) · l(3)(2) ⊗ g(3) / f(h(3)(1), l(3)(1)) ·(h(4) · l(4)) (BE7) = g(1) . f(h(1), l(1)) f g(2) / f(h(2), l(2)), h(3) · l(3) ⊗ g(3) / f(h(4), l(4)) ·(h(5) · l(5)) (BE5) = f(g(1), h(1))f(g(2) · h(2), l(1)) ⊗ g(3) / f(h(3), l(2)) ·(h(4) · l(3)) = f(g(1), h(1))f(g(2) · h(2), l(1)) ⊗ g(3) / f(h(3)(1), l(2)(1)) ·(h(3)(2) · l(2)(2)) (BE1) = f(g(1), h(1))f(g(2) · h(2), l(1)) ⊗ (g(3) · h(3)) · l(2) as needed and the proof is now ﬁnished.

Remark 2.5.5 As it was recently noticed in [116], at the level of bialgebras the uniﬁed product appears as a special case of the crossed product introduced in [32], [33] by considering (with the notations in [116]) R : H ⊗ A → A ⊗ H and σ : H ⊗ H → A ⊗ H as follows:

R(h ⊗ a) = h(1) . a(1) ⊗ h(2) / a(2)

σ(h ⊗ g) = f(h(1), g(1)) ⊗ h(2) · g(2) However, we introduce and study the uniﬁed product in [12] and [15] as an answer to the restricted extending structures problem for Hopf algebras and, moreover, we provide properties of it and classiﬁcation results which are not mentioned in [32], [33].

76 2.5. BIALGEBRA EXTENDING STRUCTURES AND UNIFIED PRODUCTS

Examples 2.5.6 1. Let A be a bialgebra and Ω(A) = H, /, ., f an extending datum of A such that the cocycle f is trivial, that is f(g, h) = εH (g)εH (h)1A, for all g, h ∈ H. Then Ω(A) = H, /, ., f is a bialgebra extending structure of A if and only if H is a bialgebra and (A, H, /, .) is a matched pair of bialgebras. In this case, the associated uniﬁed product A n H = A ./ H is the bicrossed product of bialgebras constructed in (2.33). Conversely, a matched pair of bialgebras can be deformed using a coalgebra lazy cocycle in order to obtain a bialgebra extending structure as follows. Let (A, H, /, .) be a matched pair of bialgebras such that A has antipode SA and u : H → A a coalgebra lazy 1-cocycle in the sense of Deﬁnition 2.6.3 such that h / u(g) = hεH (g), for all h ∈ H and g ∈ G. Then Ω(A) = H, /, .0, f 0 is a bialgebra extending structure of A, where .0 and f 0 are given by

0 h . c = u(h(1))(h(2) . c(1))SA u h(3) / c(2)

0 f (h, g) = u(h(1))(h(2) . u(g(1)))SA u h(3)g(2) for all h, g ∈ H and c ∈ A. 2. Let A be a bialgebra and Ω(A) = H, /, ., f an extending datum of A such that the action / is trivial, that is h / a = εA(a)h, for all h ∈ H and a ∈ A. Then Ω(A) = H, /, ., f is a bialgebra extending structure of A if and only if H is an usual bialgebra and the following compatibility conditions are fulﬁlled:

(a) The twisted module condition (2.27) and the cocycle condition (2.28) hold;

(b) g . (ab) = (g(1) . a)(g(2) . b)

(d) g(1)h(1) ⊗ f(g(2), h(2)) = g(2)h(2) ⊗ f(g(1), h(1)) for all g, h ∈ H and a, b ∈ A.

In this case, the associated uniﬁed product A n H = A#f H is the crossed product constructed in (2.26). In particular, if A is a bialgebra, the crossed product A#f H is a bialgebra with the coalgebra structure given by the tensor product of coalgebras if and only if the compatibility conditions (c) and (d) above hold.

Let A be a bialgebra and Ω(A) = H, /, ., f a bialgebra extending structure of A. Then iA : A → AnH, iA(a) = an1H , for all a ∈ A is an injective bialgebra map, iH : H → AnH, iH (h) = 1A ⊗ h, for all h ∈ H is an injective coalgebra map and

u : A ⊗ H → A n H, u(a ⊗ h) = iA(a) • iH (h) = (a n 1H ) • (1A n h) = a n h for all a ∈ A and h ∈ H is bijective, i.e. the uniﬁed product A n H factorizes through A and H. The next theorem shows the converse of this remark: any bialgebra E that factorizes through a subbialgebra of A and a subcoalgebra H is isomorphic to a uniﬁed product. In order to avoid complicated computations we use the following elementary remark:

77 CHAPTER 2. UNIFIED PRODUCTS

Lemma 2.5.7 Let E be a bialgebra, L a coalgebra and u : L → E an isomorphism of coalgebras. Then there exists a unique algebra structure on L such that u : L → E is an isomorphism of bialgebras given by:

0 −1 0 −1 l · l := u (u(l)u(l )), 1L := u (1E) 0 for all l, l ∈ L. Furthermore, if E has an antipode SE, then L is a Hopf algebra with the −1 antipode SL := u ◦ SE ◦ u.

Proof: Straightforward: the algebra structure on L is obtained by transfering the algebra structure from E via the isomorphism of coalgebras u. The multiplication on L is a coalgebra map since it is a composition of coalgebra maps.

Theorem 2.5.8 Let E be a bialgebra, A ⊆ E a subbialgebra, H ⊆ E a subcoalgebra such that 1E ∈ H and the multiplication map u : A ⊗ H → E, u(a ⊗ h) = ah, for all a ∈ A, h ∈ H is bijective. Then, there exists Ω(A) = (H, /, ., f) a bialgebra extending structure of A such that u : A n H → E, u(a n h) = ah is an isomorphism of bialgebras. Furthermore, if E is a Hopf algebra then A n H is a Hopf algebra.

Proof: Since E is a bialgebra, the multiplication mE : E ⊗ E → E is a coalgebra map. Thus u : A ⊗ H → E is in fact an isomorphism of coalgebras, with its inverse u−1 : E → A ⊗ H which is also a coalgebra map. The k-linear map µ : H ⊗ A → A ⊗ H, µ(h ⊗ a) := u−1(ha) for all h ∈ H and a ∈ A is a coalgebra map as a composition of coalgebra maps. We deﬁne the actions ., / by the formulas:

. : H ⊗ A → A, . := (Id ⊗ εH ) ◦ µ (2.46)

/ : H ⊗ A → H,/ := (εA ⊗ Id) ◦ µ (2.47) They are coalgebra maps as compositions of coalgebra maps. Moreover, the normalization conditions (2.36) and (2.37) are trivially fulﬁlled. More explicitly, . and / are given as follows: let P h ∈ H and c ∈ A. Since u is a bijective map, there exists an unique element j αj ⊗lj ∈ A⊗H P such that hc = j αjlj. Then: X X h . c = αjεH (lj), h / c = εA(αj)lj j j Next we construct the coalgebra maps f : H ⊗ H → A and · : H ⊗ H → H. The k-linear map ν : H ⊗ H → A ⊗ H, ν(h ⊗ g) := u−1(hg) for all h, g ∈ H is a coalgebra map as a composition of coalgebra maps. We deﬁne:

f : H ⊗ H → A, f := (Id ⊗ εH ) ◦ ν (2.48)

· : H ⊗ H → H, · := (εA ⊗ Id) ◦ ν (2.49)

78 2.5. BIALGEBRA EXTENDING STRUCTURES AND UNIFIED PRODUCTS

They are coalgebra maps as compositions of coalgebra maps. The normalization conditions 1E · h = h · 1E = h and f(h, 1E) = f(1E, h) = εH (h)1A, for all h ∈ H are trivially fulfilled. In order to prove that Ω(A) = (H, /, ., f, ·) is a bialgebra extending structure of A we use Lemma 2.5.7 and then Theorem 2.5.4: the unique algebra structure that can be defined on A⊗H such that u becomes an isomorphism of bialgebras is given by: (a ⊗ h) • (c ⊗ g) = u−1u(a ⊗ h)u(c ⊗ g) = u−1(ahcg) This algebra structure on A ⊗ H coincides with the one given by (2.38) on a unified product if and only if −1 u (ahcg) = a(h(1) . c(1))f(h(2) / c(2), g(1)) ⊗ (h(3) / c(3)) · g(2) Since u is a bijective map the above formula holds if and only if: hcg = (h(1) . c(1))f(h(2) / c(2), g(1)) (h(3) / c(3)) · g(2) (2.50) holds for all c ∈ A and h, g ∈ H. Therefore, the proof is finished if we prove that the relation (2.50) holds in the bialgebra E. Let c ∈ A and h, g ∈ H. Then there exists an unique element Pn j=1 αj ⊗ lj ∈ A ⊗ H such that: n X hc = αjlj (2.51) j=1 Pn Pn Hence h . c = j=1 εH (lj)αj and h / c = j=1 εA(αj)lj. Moreover, for any j = 1, ··· , n Pm there exists an unique element i=1 Aji ⊗ Zi ∈ A ⊗ H such that: m X ljg = AjiZi (2.52) i=1 Using relations (2.48) and (2.49) we obtain: m m X X f(lj, g) = εH (Zi)Aji, lj · g = εA(Aji)Zi (2.53) i=1 i=1 and m,n X hcg = αjAjiZi (2.54) i,j=1

In what follows we use the fact that mE, . and / are coalgebra maps. For example, by applying ∆ to the relation (2.51) we obtain: n X h(1) . c(1) ⊗ h(2) / c(2) ⊗ h(3) / c(3) = εH (lj(1))αj(1) ⊗ εA(αj(2))lj(2) ⊗ εA(αj(3))lj(3) j=1 n X = εH (lj(1))αj ⊗ lj(2) ⊗ lj(3) j=1 n X = αj ⊗ lj(1) ⊗ lj(2) j=1

79 CHAPTER 2. UNIFIED PRODUCTS

Thus, we have:

n X h(1) . c(1) ⊗ h(2) / c(2) ⊗ h(3) / c(3) = αj ⊗ lj(1) ⊗ lj(2) (2.55) j=1

Moreover, by applying ∆ to the relation (2.52) and using the relation (2.53) we obtain:

m m X X f lj(1), g(1) ⊗lj(1) · g(2) = εH (Zi(1) )Aji(1) ⊗ εA(Aji(2) )Zi(2) = Aji ⊗ Zi (2.56) i=1 i=1

We denote by RHS the right hand side of (2.50). Then:

n (2.55) X RHS = αjf lj(1), g(1) lj(2) · g(2) j=1 m,n (2.56) X = αjAjiZi i,j=1 (2.54) = hcg

−1 Thus the relation (2.50) holds true and the proof is now ﬁnished since u (1E) = 1A ⊗ 1E. We use Theorem 2.5.4 in order to obtain that Ω(A) = (H, /, ., f, ·) is a bialgebra extending structure of A. Moreover, if E is a Hopf algebra then A n H is also a Hopf algebra with the −1 antipode given by SAnH = u ◦ SE ◦ u according to Lemma 2.5.7.

Next we construct an antipode for the uniﬁed product A n H.

Proposition 2.5.9 Let A be a Hopf algebra with an antipode SA and Ω(A) = (H, /, ., f) a bialgebra extending structure of A such that there exists an antimorphism of coalgebras SH : H → H such that

h(1) · SH (h(2)) = SH (h(1)) · h(2) = εH (h)1H (2.57) for all h ∈ H. Then the uniﬁed product AnH is a Hopf algebra with the antipode S : AnH → A n H given by:

S(a n g) := SA[f SH (g(2)), g(3) ] n SH (g(1)) • SA(a) n 1H (2.58) for all a ∈ A and g ∈ H.

80 2.5. BIALGEBRA EXTENDING STRUCTURES AND UNIFIED PRODUCTS

Proof: Let a n g ∈ A n H. Since the multiplication • on A n H is associative we have:

S(a(1) n g(1)) • (a(2) n g(2)) = = SA[f SH (g(2)), g(3) ] n SH (g(1)) • SA(a(1)) n 1H •(a(2) n g(4)) (2.41) = SA[f SH (g(2)), g(3) ] n SH (g(1)) • SA(a(1))a(2) n g(4) = εA(a) SA[f SH (g(2)), g(3) ] n SH (g(1)) •(1A n g(4)) (2.42) = εA(a)SA f SH (g(2)), g(3) f SH (g(1))(1), g(4)(1) nSH (g(1))(2) · g(4)(2) = εA(a)SA f SH (g(1)(2)), g(2) f SH (g(1)(1))(1), g(3)(1) nSH (g(1)(1))(2) · g(3)(2) = εA(a)SA f SH (g(1))(1), g(2) f SH (g(1))(2)(1), g(3)(1) nSH (g(1))(2)(2) · g(3)(2) = εA(a)SA f SH (g(1))(1), g(2) f SH (g(1))(2), g(3)(1) nSH (g(1))(3) · g(3)(2) = εA(a)SA f SH (g(1))(1), g(2)(1) f SH (g(1))(2), g(2)(2)(1) nSH (g(1))(3) · g(2)(2)(2) = εA(a)SA f SH (g(1))(1), g(2)(1) f SH (g(1))(2), g(2)(2) nSH (g(1))(3) · g(2)(3) = εA(a)SA f SH (g(1))(1)(1), g(2)(1)(1) f SH (g(1))(1)(2), g(2)(1)(2) n

SH (g(1))(2) · g(2)(2) (2.3) = εA(a)SA f SH (g(1))(1), g(2)(1) (1) f SH (g(1))(1), g(2)(1) (2)nSH (g(1))(2) · g(2)(2) = εA(a)εA f SH (g(1))(1), g(2)(1) 1A n SH (g(1))(2) · g(2)(2) (2.3) = εA(a)εH SH (g(1))(1) εH g(2)(1) nSH (g(1))(2) · g(2)(2)

= εA(a)1A n SH (g(1)) · g(2) = εA(a)εH (g)1A n 1H

Thus S is a left inverse of the identity in the convolution algebra Hom(AnH,AnH). By similar computations one can show that S is also a right inverse of the identity, thus is an antipode of A n H.

In Proposition 2.5.9 we imposed the condition for SH to be a coalgebra antimorphism because the algebra structure on H is not an associative one and for this reason a k-linear map SH which satisﬁes the antipode condition (2.57) is not necessarily a coalgebra antimorphism as in the classical case of Hopf algebras.

81 CHAPTER 2. UNIFIED PRODUCTS

2.6 The classiﬁcation of uniﬁed products

In this section we prove the classification theorem for unified products: as a special case, a classification theorem for bicrossed products of Hopf algebras is obtained. Our view point is inspired from Schreier’s classification theorem for extensions of an abelian group K by a group Q [133, Theorem 7.34]: they are classified by the second cohomology group H2(Q, K). Let ϕ : G → G0 be a morphism between two extensions of a group K by a group Q, i.e. ϕ is a morphism of groups such that the diagram

i π k[K] / k[G] / k[Q]

Id ϕ Id i0 π0 k[K] / k[G0] / k[Q] is commutative (we wrote the diagram for the induced morphism for group algebras). Then ϕ is an isomorphism [133, Theorem 7.32]. Now, the left hand square of the diagram is commutative if and only if ϕ is a left k[K]-module map while the right hand square of the diagram is commutative if and only if ϕ is a morphism of right k[Q]-comodules. This motivates the way of considering the classification of unified products up to an isomorphism of Hopf algebras that is also a left A-module map and a right H-comodule map. Let Ω(A) = H, /, ., f be a bialgebra extending structure of A. The unified product A n H is a right H-comodule via the coaction a n h 7→ a n h(1) ⊗ h(2), for all a ∈ A and h ∈ H and a left A-module via the restriction of scalars map iA : A → A n H. From now on the Hopf algebra structure on A and the coalgebra structure on H will be set. First, we need the following.

Lemma 2.6.1 Let A be a Hopf algebra, Ω(A) = H, /, ., f and Ω0(A) = H,/0,.0, f 0 two 0 Hopf algebra extending structures of A. Then a k-linear map ϕ : A n H → A n H is a left A-module, a right H-comodule and a coalgebra morphism if and only if there exists a unique morphism of coalgebras u : H → A such that

h(1) ⊗ u(h(2)) = h(2) ⊗ u(h(1)) (2.59) for all h ∈ H and ϕ is given by

0 ϕ(a n h) = au(h(1)) n h(2) (2.60)

0 for all a ∈ A and h ∈ H. Furthermore, any such a morphism ϕ : A n H → A n H is an isomorphism with the inverse given by

0 0 ψ : A n H → A n H, ψ(a n h) = aSA u(h(1)) nh(2) for all a ∈ A and h ∈ H.

82 2.6. THE CLASSIFICATION OF UNIFIED PRODUCTS

0 Proof: Let ϕ : A n H → A n H be a left A-module, a right H-comodule and a coalgebra P A H morphism. We shall adopt the notation ϕ(1A n h) = h ⊗ h ∈ A ⊗ H, for all h ∈ H. Since ϕ is a left A-module map we have

X A H ϕ(a n h) = aϕ(1A n h) = a h ⊗ h for all a ∈ A and h ∈ H. As ϕ is also a right H-comodule map we have:

X A H H ah ⊗ (h )(1) ⊗ (h )(2) = ϕ(a n h(1)) ⊗ h(2)

By applying εH on the second position of the above identity we obtain:

X A H ϕ(a n h) = a(h(1)) εH ((h(1)) ) ⊗ h(2) for all a ∈ A and h ∈ H. Now, if we deﬁne u : H → A by:

X A H u(h) = (Id ⊗ εH ) ◦ ϕ(1A n h) = h εH (h ) for all h ∈ H, it follows that (2.60) holds. We shall prove now that ϕ given by (2.60) is a coalgebra map if and only if u is a coalgebra map and (2.59) holds. First we observe that

εAn0H ◦ ϕ = εAnH if and only if εA ◦ u = εH . Now, the fact that ϕ is comultiplicative is equivalent to:

u(h(1))(1) ⊗ h(2) ⊗ u(h(1))(2) ⊗ h(3) = u(h(1)) ⊗ h(2) ⊗ u(h(3)) ⊗ h(4) (2.61) for all h ∈ H. By applying Id ⊗ εH ⊗ Id ⊗ εH to this relation we obtain that u is a coalgebra map; using this fact and then applying εA ⊗ Id ⊗ Id ⊗ εH in relation (2.61) we obtain relation (2.59). Conversely, if u is a coalgebra map such that relation (2.59) holds, then (2.61) follows straightforward, i.e. ϕ is a coalgebra map. The fact that ψ is an inverse for φ is also straightforward.

Remark 2.6.2 At this point we should remark the perfect similarity with the theory of extensions 0 from the groups case. If ϕ : A n H → A n H is a left A-module, a right H-comodule and a coalgebra morphism between two uniﬁed products then the following diagram

iA πH A / A ./ H / H

IdA ϕ IdH

iA πH A / A ./0 H / H is commutative and ϕ is an isomorphism.

1 For A = k and a Hopf algebra H the group Hl (H, k) of all unitary algebra maps u : H → k satisfying the compatibility condition (2.62) below was called in [34, Definition 1.1] the first lazy cohomology group of H with coefficients in k. We shall now define the coalgebra version of lazy 1-cocyles.

83 CHAPTER 2. UNIFIED PRODUCTS

Deﬁnition 2.6.3 Let A be a Hopf algebra and H a coalgebra, unitary not necessarily associative algebra. A morphism of coalgebras u : H → A is called a coalgebra lazy 1-cocyle if u(1H ) = 1A and the following compatibility holds:

h(1) ⊗ u(h(2)) = h(2) ⊗ u(h(1)) (2.62) 1 for all h ∈ H. We denote by Hl,c(H,A) the group of all coalgebra lazy 1-cocyles of H with coefﬁcients in A.

1 Hl,c(H,A) is a group with respect to the convolution product. We have to prove that if u and 1 1 v ∈ Hl,c(H,A), then u ∗ v ∈ Hl,c(H,A). Indeed, is straightforward to prove that u ∗ v satisfy (2.62). Let us show that u ∗ v is a morphism of coalgebras. First, if we apply v on the ﬁrst position in (2.62) we obtain v(h(1)) ⊗ u(h(2)) = v(h(2)) ⊗ u(h(1)), for all h ∈ H. Using this relation we obtain: ∆A u(h(1))v(h(2)) = u(h(1))v(h(3)) ⊗ u(h(2))v(h(4))

= u(h(1))v(h(2)) ⊗ u(h(3))v(h(4))

= u ∗ v(h(1)) ⊗ u ∗ v(h(2)) for all h ∈ H, hence u ∗ v is also a coalgebra map. The main theorem of this section now follows:

Theorem 2.6.4 Let A be a Hopf algebra, Ω(A) = H, /, ., f and Ω0(A) = H,/0,.0, f 0 0 two Hopf algebra extending structures of A. Then there exists ϕ : A n H → A n H a left A-module, a right H-comodule and a Hopf algebra map if and only if /0 = / and there exists a 1 coalgebra lazy 1-cocyle u ∈ Hl,c(H,A) such that:

0 h . c = u(h(1))(h(2) . c(1))SA u h(3) / c(2) (2.63)

0 0 f (h, g) = u(h(1))(h(2) . u(g(1)))f(h(3) / u(g(2)), g(3))SA u h(4) · g(4) (2.64) 0 h · g = h / u(g(1)) · g(2) (2.65) for all h, g ∈ H and c ∈ A. In this case ϕ is given by (2.60) and it is an isomorphism.

0 Proof: We already proved in Lemma 2.6.1 that ϕ : A n H → A n H is a left A-module, a right 0 H-comodule and a coalgebra map if and only if ϕ(a n h) = au(h(1)) n h(2), for all a ∈ A, h ∈ H and for a unique coalgebra map u : H → A such that the (2.62) holds. Of course, 0 ϕ(1A n 1H ) = 1A n 1H if and only if u is unitary. Moreover, as u is a morphism of coalgebras −1 it is invertible in convolution with the inverse u = SA ◦ u. In what follows we shall prove, in the hypothesis that ϕ is a coalgebra map and u is unitary, that ϕ is an algebra map (thus a map of bialgebras) if and only if /0 = / and the compatibility conditions (2.63)-(2.65) hold. By a straightforward computation we can show that ϕ is an algebra map if and only if 0 0 0 0 0 0 0 (C)(h(1) . c(1))f h(2) / c(2), g(1) u (h(3) / c(3)) · g(2) n(h(4) / c(4)) · g(3) = = u(h(1)) h(2) . c(1)u(g(1)) f h(3) / c(2)u(g(2)), g(4) n h(4) / c(3)u(g(3)) ·g(5)

84 2.6. THE CLASSIFICATION OF UNIFIED PRODUCTS for all h, g ∈ H and c ∈ A. We shall prove that the compatibility (C) is equivalent to (2.63)- (2.65). 0 Indeed, by considering g = 1H in (C) and then by applying εA ⊗ Id we obtain h / c = h / c, for all h ∈ H and c ∈ A. If we consider again g = 1H we obtain, after applying first Id ⊗ εH and then inverting u, that (2.63) holds. Relation (2.65) is obtained by considering c = 1A in (C), applying Id ⊗ εH and finally inverting u. To end with, relation (2.65) follows by considering c = 1A and by applying εA ⊗ Id in (C). Conversely, suppose that h/0 c = h/c, for all h ∈ H and c ∈ A and there exists a coalgebra lazy 1-cocyle u such that relations (2.63)-(2.65) are fulfilled. We then have (we denote by LHS the left hand side of (C)): LHS = u(h(1)(1))(h(1)(2) . c(1)(1))SA u h(1)(3) / c(1)(2) u(h(2)(1) / c(2)(1)) (h(2)(2) / c(2)(2)) . u(g(1)(1)) f (h(2)(3) / c(2)(3)) / u(g(1)(2)), g(1)(3) 0 0 0 SA u h(2)(4) / c(2)(4) · g(1)(4) u (h(3) / c(3)) · g(2) n(h(4) / c(4)) · g(3) = u(h(1))(h(2) . c(1))SA u h(3) / c(2) u(h(4) / c(3)) (h(5) / c(4)) . u(g(1))

0 0 f (h(6) / c(5)) / u(g(2)), g(3) SA u h(7) / c(6) · g(4) u (h(8) / c(7)) · g(5) 0 n(h(9) / c(8)) · g(6) = u(h(1))(h(2) . c(1)) (h(3) / c(2)) . u(g(1)) f (h(4) / c(3)) / u(g(2)), g(3) 0 n(h(5) / c(4)) · g(4) (2d) = u(h(1)) h(2) . c(1)u(g(1)) f (h(3) / c(2)) / u(g(2)), g(3) 0 n(h(4) / c(3)) · g(4) (2.65) = u(h(1)) h(2) . c(1)u(g(1)) f (h(3) / c(2)) / u(g(2)), g(3)

n(h(4) / c(3)u(g(4))) · g(5) (2.62) = u(h(1)) h(2) . c(1)u(g(1)) f (h(3) / c(2)) / u(g(2)), g(4)

n(h(4) / c(3)u(g(3))) · g(5) where in the first equality we used relations (2.63)-(2.65) while the third equality holds by using the antipode conditions and the fact that u is a coalgebra map. Thus (C) holds and the proof is now finished. Even if for the classification problem we only set the Hopf algebra structure of A and the coalgebra structure of H, Theorem 2.6.4 tells us that we can set also the coalgebra map / : H⊗A → H. We shall phrase Theorem 2.6.4 as a description of the skeleton for the category C(A, H, /) defined below.

Let A be a Hopf algebra, H a coalgebra with a ﬁxed group-like element 1H ∈ H and / : H ⊗ A → H a morphism of coalgebras. Let ES(A, H, /) be the set of all triples (·, ., f) such

85 CHAPTER 2. UNIFIED PRODUCTS

that (H, 1X , ·), /, ., f is a Hopf algebra extending structure of A. The next definition is the Hopf algebra version for unified product [133, Definition 7.31] given for extensions of groups.

Deﬁnition 2.6.5 Two elements (·, ., f), (·0,.0, f 0) of ES(A, H, /) are called cohomologous and 0 0 0 1 we denote this by (·, ., f) ≈ (· ,. , f ) if there exists a coalgebra lazy 1-cocyle u ∈ Hl,c(H,A) such that the compatibility conditions (2.63)-(2.65) are fulﬁlled.

0 0 0 0 It follows from Theorem 2.6.4 that (·, ., f) ≈ (· ,. , f ) if and only if there exists ϕ : A n H → A n H a left A-module, a right H-comodule and a Hopf algebra map. Moreover, from 0 Lemma 2.6.1 we obtain that any such map ϕ : A n H → A n H is an isomorphism, thus ≈ 2 is an equivalence relation on the set ES(A, H, /). We denote by Hl,c(H, A, /) the quotient set ES(A, H, /)/ ≈. Let C(A, H, /) be the category whose class of objects is the set ES(A, H, /). A morphism 0 0 0 ϕ : ·, ., f → ·,. , f in C(A, H, /) is a Hopf algebra morphism ϕ : A n H → A n H that is a left A-module and a right H-comodule map. Thus we obtain the categorical version of Theorem 2.6.4:

Corollary 2.6.6 (Schreier theorem for uniﬁed products) Let A be a Hopf algebra, H a coalgebra with a group-like element 1H and / : H ⊗ A → H a morphism of coalgebras. There exists a bijection between the set of objects of the skeleton of the category C(A, H, /) and the quotient 2 set Hl,c(H, A, /).

2 Hl,c(H, A, /) is for the classification of the unified products the counterpart of the second cohomology group for the classification of an extension of an abelian group by a group [133, Theorem 7.34]. We can apply Theorem 2.6.4 to obtain classification theorems for various special cases of the unified products: for instance, Doi’s results on the classification of crossed products ([61]) is obtain as a special case if we let /0 = / be the trivial actions. Now, we shall indicate the classification of bicrossed product of Hopf algebras.

Corollary 2.6.7 (Schreier theorem for bicrossed products) Let A and H be two Hopf algebras and A, H, /, ., A, H, /0,.0 two matched pairs of Hopf algebras. Then A ./ H =∼ A ./0 H (isomorphism of Hopf algebras, left A-modules and right H-comodules) if and only if /0 = / 1 and there exists a coalgebra lazy 1-cocyle u ∈ Hl,c(H,A) such that:

0 h . c = u(h(1))(h(2) . c(1))SA u h(3) / c(2) u(h(1))(h(2) . u(g(1)))SA u h(3)g(2) = εH (g)εH (h)1A

h / u(g) = h εH (g) for all h, g ∈ H and c ∈ A.

86 2.6. THE CLASSIFICATION OF UNIFIED PRODUCTS

Proof: We apply Theorem 2.6.4 for the case when f and f 0 are the trivial cocycles. As the multiplication on the algebra H is the same (i.e. · = ·0), the condition (2.65) in Theorem 2.6.4 takes the equivalent form h / u(g) = hεH (g), for all h, g ∈ H.

The construction of unified products is a challenging problem considering the number of compatibilities that need to be fulfilled. In particular, an example of an unified product which is neither a crossed product nor a bicrossed product is interesting in the picture. We provide such ∼ an example below: it is a Hopf algebra k[A4] n k[S] = k[A6], where An is the alternating group on a set with n elements and S is a set with thirty elements.

Example 2.6.8 Let G be a group and (X, 1X ) a pointed set. We consider the group Hopf algebra A := k[G] and the group-like coalgebra H := k[X]. We note that coalgebra morphisms between two group-like coalgebras are in one to one correspondence with the maps between the corresponding sets. Thus, any bialgebra extending structure (k[X], /, ., f) of the Hopf algebra k[G] is induced by an extending structure (X,/0,.0, f 0) of the group G. Moreover there exists a canonical isomorphism of bialgebras ∼ k[G] n k[X] = k[G n X] where G n X is the uniﬁed product at the level of groups. This generalizes the fact that a bicrossed product of two group Hopf algebras is isomorphic to the group Hopf algebra of the bicrossed product of the corresponding groups. The same type of isomorphism holds also for crossed products of Hopf algebras between two group Hopf algebras.

Now, let A6 be the alternating group on a set with six elements. A6 is the simple group of smallest order that cannot be written as a bicrossed product of two proper subgroups ([146]). Being a simple group it can not be written neither as a crossed product of two proper subgroups. On the other hand, A6 can be written as an uniﬁed product between any of its subgroups and an extending structure. For instance, we can write ∼ A6 = A4 n S for an extending structure S, 1S, α, β, f, ∗, i of A4, where S is a set of representatives for the right cosets of A4 in A6 with 30 elements such that 1 ∈ S. Thus there exists an example of ∼ an uniﬁed product for Hopf algebras k[A4] n k[S] = k[A6] which is neither a crossed product nor a bicrossed product of two Hopf algebras.

Two general methods for constructing uniﬁed products are proposed in [15]. One of them con- structs an uniﬁed product starting with a minimal set of data: a Hopf algebra A, a unitary not necessarily associative bialgebra H which is a right A-module coalgebra and a unitary coalgebra map γ : H → A satisfying four technical compatibility conditions ([15, Theorem 2.9]). This will be the topic of our next section.

87 CHAPTER 2. UNIFIED PRODUCTS

2.7 Uniﬁed products and split extensions of Hopf algebras

For any bialgebra map i : A → E, E will be viewed as a left A-module via i, that is a·x = i(a)x, for all a ∈ A and x ∈ E. We shall adopt a deﬁnition from [21] due in the context of Hopf algebra maps.

Deﬁnition 2.7.1 Let A and E be two bialgebras. A coalgebra map π : E → A is called normal if the space {x ∈ E | π(x(1)) ⊗ x(2) = 1A ⊗ x} is a subcoalgebra of E.

Let G be a ﬁnite group, H ≤ G a subgroup of G and k[G]∗ the Hopf algebra of functions on G. Then the restriction morphism k[G]∗ → k[H]∗ is a normal morphism if and only if H is a normal subgroup of G ([21]). The main properties of the bialgebra extension A ⊂ A n H are given by the following:

Proposition 2.7.2 Let A be a bialgebra, Ω(A) = H, /, ., f a bialgebra extending structure of A and the k-linear maps:

iA : A → A n H, iA(a) = a n 1H , πA : A n H → A, πA(a n h) = εH (h)a for all a ∈ A, h ∈ H. Then:

1. iA is a bialgebra map, πA is a normal left A-module coalgebra morphism and πA ◦ iA = IdA.

2. πA is a right A-module map if and only if . is the trivial action.

3. πA is a bialgebra map if and only if . and f are the trivial maps, i.e. the uniﬁed product A n H = A#H, the right version of the smash product of bialgebras.

Proof: (1) The fact that iA is a bialgebra map and πA is a coalgebra map is straightforward. We show that πA is normal. Indeed, the subspace nX X X o ai n hi ∈ A n H | ai(1) εH (hi(1) ) ⊗ ai(2) n hi(2) = 1A ⊗ ai n hi i i i is a subcoalgebra in A n H since it can be identiﬁed with 1A n H: more precisely, applying P εA on the second position in the equality from the above subspace we obtain that i ai n hi = P 1A n i εA(ai)hi ∈ 1A n H, as needed. On the other hand πA a · (c n h) = πA iA(a) • (c n h) = πA(ac n h) = ac εH (h) = a πA(c n h) for all a, c ∈ A and h ∈ H, i.e. πA is a left A-module map. (2) and (3) are proven by a straightforward computation.

88 2.7. UNIFIED PRODUCTS AND SPLIT EXTENSIONS OF HOPF ALGEBRAS

The next Theorem gives the converse of Proposition 2.7.2 (1) and the generalization of [105, Theorem 4.1]. Our proof is based on the factorization Theorem 2.5.8 and the fundamental theorem of Hopf-modules.

Theorem 2.7.3 Let i : A → E be a Hopf algebra morphism such that there exists π : E → A a normal left A-module coalgebra morphism for which π ◦ i = IdA. Let

H := {x ∈ E | π(x(1)) ⊗ x(2) = 1A ⊗ x} Then there exists a bialgebra extending structure Ω(A) = H, /, ., f of A, where the multiplication on H, the actions ., / and the cocycle f are given by the formulas: h · g := i SA π(h(1)g(1)) h(2)g(2), f(h, g) := π(hg) h / a := i SA π(h(1)i(a(1))) h(2)i(a(2)), h . a := π hi(a) for all h, g ∈ H, a ∈ A such that

ϕ : A n H → E, ϕ(a n h) = i(a)h for all a ∈ A and h ∈ H is an isomorphism of Hopf algebras.

Proof: There are two possible ways to prove the above theorem: the ﬁrst one is to give a standard proof in the way this type of theorems are usually proved in Hopf algebra theory, by showing that all compatibility conditions (BE1) − (BE7) hold and then to prove that ϕ is an isomorphism of Hopf algebras. We prefer however a more direct approach which relies on Theorem 2.5.8: it has the advantage of making more transparent to the reader the way we obtained the formulas which deﬁne the bialgebra extending structure Ω(A). First we observe that 1E ∈ H since π(1E) = π(i(1A)) = 1A and H is a subcoalgebra of E as π is normal. E has a structure of left-left A-Hopf module via the left A-action and the left A-coaction given by

a · x := i(a)x, ρ(x) = x<−1> ⊗ x<0> := π(x(1)) ⊗ x(2) for all a ∈ A and x ∈ E. Indeed let us prove the Hopf module compatibility condition: ρ(a · x) = ρ i(a)x = π i(a(1))x(1) ⊗i(a(2))x(2) = π a(1) · x(1) ⊗i(a(2))x(2) = a(1)π(x(1)) ⊗ a(2) · x(2)

= a(1)x<−1> ⊗ a(2) · x<0> for all a ∈ A and x ∈ E, i.e. E is a left-left A-Hopf module. We also note that H = Eco(A). It follows from the fundamental theorem of Hopf modules that the map

ϕ : A ⊗ H → E, ϕ(a ⊗ x) := a · x = i(a)x for all a ∈ A and h ∈ H is an isomorphism of vector spaces with the inverse given by

−1 ϕ (x) := x<−2> ⊗ SA(x<−1>) · x<0> = π(x(1)) ⊗ i SA π(x(2)) x(3) (2.66)

89 CHAPTER 2. UNIFIED PRODUCTS for all x ∈ E. Thus E is a Hopf algebra that factorizes through i(A) =∼ A and the subcoalgebra H. Now, from Theorem 2.5.8 we obtain that there exists a bialgebra extending structure Ω(A) = H, /, ., f of A such that ϕ : A n H → E, ϕ(a n h) = i(a)h, for all a ∈ A and h ∈ H is an isomorphism of Hopf algebras. Using (2.66) the actions ., / given by the formulas (2.46), (2.47) take the explicit form:

−1 h . a = (IdA ⊗ εH )ϕ (hi(a)) = π h(1)i(a(1)) εH (h(2))εA(a(2))εH (h(3))εA(a(3)) = π hi(a) and

−1 h / a = (εA ⊗ IdE)ϕ (hi(a)) = i SA π(h(1)i(a(1))) h(2)i(a(2)) for all h ∈ H and a ∈ A. Finally, using once again (2.66), the multiplication · on H and the cocycle f given by (2.48), (2.49) take the form h · g := i SA π(h(1)g(1)) h(2)g(2), f(h, g) := π(hg) for all h, g ∈ H. The proof is now ﬁnished by Theorem 2.5.8.

The next corollary covers the case in which the splitting map π is also a right A-module map. The version of Corollary 2.7.4 below in which the normality condition of the splitting map π is dropped was proved in [26, Theorem 3.64]. In this case, the input data of the construction of the product that is used is called a dual Yetter-Drinfel’d quadruple [26, Deﬁniton 3.59], and consists of a system of objects and maps satisfying eleven compatibility conditions.

Corollary 2.7.4 Let i : A → E be a Hopf algebra morphism such that there exists π : E → A a normal A-bimodule coalgebra morphism such that π ◦ i = IdA. Then there exists Ω(A) = (H, /, f) a twisted bialgebra extending structure of A such that

ϕ : A♦H → E, ϕ(a♦h) = i(a)h for all a ∈ A and h ∈ H is an isomorphism of Hopf algebras.

Proof: It follows from Theorem 2.7.3 that there exists a bialgebra extending structure Ω(A) = H, /, ., f of A such that

ϕ : A n H → E, ϕ(a n h) = i(a)h for all a ∈ A and h ∈ H is an isomorphism of Hopf algebras. We observe that for any h ∈ H = {x ∈ E | π(x(1)) ⊗ x(2) = 1A ⊗ x}, we have that π(h) = εH (h)1A. Thus, as π is also a right A-module map, the action . is given by: h . a = π hi(a) = π(h · a) = π(h)a = εH (h)a for all h ∈ H and a ∈ A. So the action . is trivial and hence the uniﬁed product A n H is a twisted product A♦H.

90 2.7. UNIFIED PRODUCTS AND SPLIT EXTENSIONS OF HOPF ALGEBRAS

The following is a simpliﬁed and more transparent version of [105, Theorem 4.1]. We use a minimal context: only the concept of normality of a morphism in the sense of [21] is used, contrary to [105, Deﬁnition 3.5] where it was taken as input data for [105, Theorem 4.1].

Corollary 2.7.5 Let π : E → A be a normal split epimorphism of Hopf algebras. Then there exists an isomorphism of Hopf algebras E =∼ A#H, for some right A-module bialgebra H, where A#H is the right version of the smash product of bialgebras.

Proof: Let i : A → E be a Hopf algebra map such that π ◦ i = IdA and

H = {x ∈ E | π(x(1)) ⊗ x(2) = 1A ⊗ x} which is a subcoalgebra in E as π is normal. In fact, as π is also an algebra map, H is a Hopf subalgebra of E. We note that π(h) = εH (h)1A, for all h ∈ H. π is also a A-bimodule map as π(a · h) = π(i(a)h) = π(i(a))π(h) = aπ(h) and π(h · a) = π(hi(a)) = π(h)a, for all a ∈ A and h ∈ H. Now we can apply Theorem 2.7.3. Thus, there exists a bialgebra extending structure Ω(A) = H, /, ., f of A such that

ϕ : A n H → E, ϕ(a n h) = i(a)h is an isomorphism of bialgebras. Moreover the action . is the trivial one as π is a right A-module map (Corollary 2.7.4). On the other hand, the cocycle f as it was defined in Theorem 2.7.3 is also the trivial one: f(h, g) = π(hg) = π(h)π(g) = εH (h)εH (g)1A, for all h, g ∈ H. Using once again the fact that π is an algebra map, the multiplication · on H as it was defined in Theorem 2.7.3 is exactly the one of E (and this fits with the fact that H is a Hopf subalgebra of E). Finally, the right action as it was defined in Theorem 2.7.3 takes the simplified form: h / a = i SA(a(1)) h a(1), for all h ∈ H and a ∈ A. Thus, the unified product A n H is the right version of the smash product of Hopf algebras from Example 2.5.6 and ϕ : A#H → E is an isomorphism of Hopf algebras.

From now on, A will be a Hopf algebra and Ω(A) = H, /, ., f a bialgebra extending structure of A. Let iA : A → AnH, iA(a) = an1H , for all a ∈ A be the canonical bialgebra morphism.

Proposition 2.7.6 Let A be a Hopf algebra and Ω(A) = H, /, ., f a bialgebra extending structure of A. Then iA : A → A n H is a split monomorphism in the category of bialgebras if and only if there exists γ : H → A a unitary coalgebra map such that

−1 h . a = γ(h(1)) a(1) γ (h(2) / a(2)) (2.67) −1 f(h, g) = γ(h(1)) γ(g(1)) γ (h(2) · g(2)) (2.68) −1 for all h, g ∈ H and a ∈ A, where γ = SA ◦ γ.

Proof: iA : A → A n H is a split monomorphism of bialgebras if and only if there exists a bialgebra map p : A n H → A such that p(a n 1) = a, for all a ∈ A. Such a Hopf algebra map p is given by: p(a n h) = p (a n 1H ) • (1A n h) = a p(1A#h)

91 CHAPTER 2. UNIFIED PRODUCTS for all a ∈ A and h ∈ H. We denote by γ : H → A, γ(h) := p(1 n h), for all h ∈ H. Hence, such a splitting map p should be given by

p = pγ : A n H → A, pγ(a n h) = aγ(h) for all a ∈ A and h ∈ H. First we note that pγ is a coalgebra map if and only if γ is a coalgebra map. Now, we prove that pγ is an algebra map if and only if γ(1H ) = 1A and the following two compatibilities are fulﬁlled: γ(h)a = h(1) . a(1) γ(h(2) / a(2)) (2.69)

γ(h)γ(g) = f(h(1), g(1)) γ(h(2) · g(2)) (2.70) for all h, g ∈ H and a ∈ A. Of course, pγ(1A n 1H ) = 1A if and only γ(1H ) = 1A. We assume now that γ(1H ) = 1A. Then, pγ is an algebra map if and only if pγ(xy) = pγ(x)pγ(y), for all x, y ∈ T := {a n 1H | a ∈ A} ∪ {1A n g | g ∈ H}, the set of generators as an algebra of A n H (see [16]). Now, for any a, c ∈ A and h ∈ H we have: pγ (a n 1H ) • (c n 1H ) = ac = pγ(a n 1H )pγ(c n 1H ) and pγ (a n 1H ) • (1A n h) = aγ(h) = pγ(a n 1H )pγ(1A n h) On the other hand, it is straightforward to see that pγ (1A n h) • (a n 1H ) = pγ(1A n h)pγ(a n 1H ) if and only if (2.69) holds and pγ (1A n h) • (1A n g) = pγ(1A n h)pγ(1A n g) if and only if (2.70) holds. Thus, we have proved that pγ is a bialgebra map if and only if γ : H → A is a unitary coalgebra map and the compatibility conditions (2.69) and (2.70) are fulﬁlled. Being a coalgebra map, γ is invertible in convolution with the inverse given by −1 γ = SA ◦ γ. We observe that (2.69) is equivalent to (2.67) while (2.70) is equivalent to (2.68) and the proof is ﬁnished.

Suppose now that we are in the setting of Proposition 2.7.6. The multiplication on the unified product A n H given by (2.38) takes the following form −1 (a n h) • (c n g) = aγ(h(1))c(1)γ(g(1))γ (h(2) / c(2)) · g(2) n(h(3) / c(3)) · g(3) (2.71) for all a, c ∈ A and h, g ∈ H, which is still very difficult to deal with. Next we shall give another equivalent description of the bialgebra structure on this special unified product in which the multiplication has a less complicated form. Let A ~ H = A ⊗ H, as a k-module with the unit 1A ~ 1H and the following structures: −1 (a ~ h) · (c ~ g) := ac(1) ~ h / c(2)γ (g(1) ·g(2) (2.72) −1 ∆A~H (a ~ h) := a(1) ~ h(2) ⊗ a(2)γ (h(1))γ(h(3)) ~ h(4) (2.73)

εA~H (a ~ h) := εA(a)εH (h) (2.74)

92 2.7. UNIFIED PRODUCTS AND SPLIT EXTENSIONS OF HOPF ALGEBRAS for all a, c ∈ A and h, g ∈ H, where we denoted a ⊗ h ∈ A ⊗ H by a ~ h. The object A ~ H introduced above is an interesting deformation of the smash product of bialgebras A#H of Example 2.5.6 that can be recovered in the case that γ : H → A is the trivial map, that is γ(h) = εH (h)1A, for all h ∈ H. If H is cocommutative then the comultiplication given by (2.73) is just the tensor product of coalgebras.

Proposition 2.7.7 Let Ω(A) = H, /, ., f be a bialgebra extending structure of a Hopf algebra A and γ : H → A be a unitary coalgebra map such that (2.67) and (2.68) hold. Then

ϕ : A n H → A ~ H, ϕ(a n h) := aγ(h(1)) ~ h(2) for all a ∈ A and h ∈ H is an isomorphism of bialgebras.

−1 Proof: The map ϕ : AnH → A⊗H is bijective with the inverse ϕ : A⊗H → AnH given −1 −1 −1 by ϕ (a ⊗ h) = aγ (h(1)) n h(2), for all a ∈ A and h ∈ H, where γ = SA ◦ γ. Below we shall use the fact that γ−1 : H → A is an antimorphism of coalgebras, i.e. ∆γ−1(h)= −1 −1 γ (h(2)) ⊗ γ (h(1)), for all h ∈ H. The unique algebra structure that can be deﬁned on the k-module A ⊗ H such that ϕ : A n H → A ⊗ H becomes an isomorphism of algebras is given by: (a ⊗ h) · (c ⊗ g) = ϕ ϕ−1(a ⊗ h) • ϕ−1(c ⊗ h)

−1 −1 = ϕ aγ (h(1)) n h(2) • cγ (g(1)) n g(2)

(2.71) −1 −1 = ϕ aγ (h(1))γ(h(2))c(1)γ (g(3))γ(g(4))

−1 −1 −1 γ [(h(3) / (c(2)γ (g(2)))) · g(5)] n [(h(4) / (c(3)γ (g(1)))) · g(6)]

−1 −1 = ϕ ac(1)γ [(h(1) / (c(2)γ (g(2)))) · g(3)]

−1 n(h(2) / (c(3)γ (g(1)))) · g(4)

−1 −1 = ac(1)γ [(h(1) / (c(2)γ (g(3)))) · g(4)] −1 −1 γ[(h(2) / (c(3)γ (g(2)))) · g(5)] ⊗ (h(3) / (c(4)γ (g(1)))) · g(6) −1 = ac(1) ⊗ h / c(2)γ (g(1) ·g(2) which is exactly the multiplication deﬁned by (2.72) on A ~ H = A ⊗ H. Thus, we have proved that ϕ : A n H → A ~ H is an isomorphism of algebras. It remains to prove that ϕ : A n H → A ~ H is also a coalgebra map. For any a ∈ A and h ∈ H we have: ∆A~H ϕ(a n h) = ∆A~H aγ(h(1)) ~ h(2) (2.73) −1 = a(1)γ(h(1)) ~ h(4) ⊗ a(2)γ(h(2))γ (h(3))γ(h(5)) ~ h(6) = a(1)γ(h(1)) ~ h(2) ⊗ a(2)γ(h(3)) ~ h(4) = ϕ(a(1) n h(1)) ⊗ ϕ(a(2) n h(2))

93 CHAPTER 2. UNIFIED PRODUCTS i.e. ϕ : A n H → A ~ H is a coalgebra map, as needed. By assumption, A n H is a bialgebra, thus A ~ H is a bialgebra and ϕ is an isomorphism of bialgebras.

Proposition 2.7.6 provides a method to construct bialgebra extending structures, and thus uniﬁed products, starting only with a right action / and a unitary coalgebra map γ : H → A.

Theorem 2.7.8 Let A be a Hopf algebra, H a unitary not necessarily associative bialgebra such that (H,/) is a right A-module coalgebra with 1A / a = εA(a)1H , for all a ∈ A. Let γ : H → A be a unitary coalgebra map and deﬁne

−1 .γ : H ⊗ A → A, h .γ a : = γ(h(1)) a(1) γ (h(2) / a(2)) (2.75) −1 fγ : H ⊗ H → A, fγ(h, g) : = γ(h(1)) γ(g(1)) γ (h(2) · g(2)) (2.76) for all h, g ∈ H and a ∈ A. Assume that the compatibility conditions (BE1), (BE3), (BE6) and (BE7) hold for .γ and fγ.

Then Ω(A) = (H, /, .γ, fγ) is a bialgebra extending structure of A and there exists an isomor- ∼ phism of bialgebras A n H = L ∗ A, where L ∗ A is the Radford biproduct for a bialgebra L in A the category AYD of Yetter-Drinfel’d modules.

−1 Proof: First we have to prove that .γ and fγ are coalgebra maps. Using the fact that γ = SA ◦ γ is an antimorphism of coalgebras we have:

−1 −1 ∆A(h .γ a) = γ(h(1))a(1)γ (h(4) / a(4)) ⊗ γ(h(2))a(2)γ (h(3) / a(3)) −1 = γ(h(1))a(1)γ (h(3) / a(3)) ⊗ h(2) .γ a(2) (BE6) −1 = γ(h(1))a(1)γ (h(2) / a(2)) ⊗ h(3) .γ a(3)

= h(1) .γ a(1) ⊗ h(2) .γ a(2) for all h ∈ H and a ∈ A, i.e. .γ is a coalgebra map. On the other hand, we have:

−1 −1 ∆A(fγ(h, g)) = γ(h(1)) γ(g(1)) γ (h(4) · g(4)) ⊗ γ(h(2)) γ(g(2)) γ (h(3) · g(3)) −1 = γ(h(1)) γ(g(1)) γ (h(3) · g(3)) ⊗ fγ(h(2), g(2)) (BE7) = fγ(h(1), g(1)) ⊗ fγ(h(2), g(2))

= (fγ ⊗ fγ)∆H⊗H (h ⊗ g) for all h, g ∈ H, that is fγ is a coalgebra map.

It remains to prove that the compatibility conditions (BE2), (BE4) and (BE5) hold for .γ and fγ. For a, b ∈ A and g ∈ H we have:

−1 g .γ (ab) = γ(g(1))a(1)b(1)γ g(2) / (a(2)b(2)) −1 −1 = γ(g(1))a(1)γ (g(2) / a(2))γ(g(3) / a(3))b(1)γ g(4) / (a(4)b(2))

= (g(1) .γ a(1))[(g(2) / a(2)) .γ b]

94 2.7. UNIFIED PRODUCTS AND SPLIT EXTENSIONS OF HOPF ALGEBRAS i.e. (BE2) holds. We denote by LHS (resp. RHS) the left (resp. right) hand side of (BE4). We have:

−1 LHS = γ(g(1))(h(1) .γ a(1))γ g(2) / (h(2) .γ a(2)) γ g(3) / (h(3) .γ a(3))

−1 γ(h(5) / a(5))γ g(4) / (h(4) .γ a(4)) ·(h(6) / a(6))

−1 = γ(g(1))(h(1) .γ a(1))γ(h(3) / a(3))γ g(2) / (h(2) .γ a(2)) ·(h(4) / a(4))

(BE6) −1 = γ(g(1))(h(1) .γ a(1))γ(h(2) / a(2))γ g(2) / (h(3) .γ a(3)) ·(h(4) / a(4))

(BE3) −1 = γ(g(1))(h(1) .γ a(1))γ(h(2) / a(2))γ g(2) · h(3) / a(3)

(2.75) −1 = γ(g(1))γ(h(1))a(1)γ g(2) · h(2) / a(2) = RHS for all a ∈ A, h, g ∈ H, i.e. (BE4) holds. Now, for g, h and l ∈ H the right hand side of (BE5) takes the following form:

−1 RHS = γ(g(1))γ(h(1))γ(l(1))γ (g(2) · h(2)) · l(2) while the left hand side of (BE5) is

−1 LHS = γ(g(1))fγ(h(1), l(1))γ(h(3) · l(3))γ g(2) / fγ(h(2), l(2)) ·(h(4) · l(4))

(BE7) −1 = γ(g(1))fγ(h(1), l(1))γ(h(2) · l(2))γ g(2) / fγ(h(3), l(3)) ·(h(4) · l(4))

(BE1) −1 = γ(g(1))fγ(h(1), l(1))γ(h(2) · l(2))γ (g(2) · h(3)) · l(3) (2.76) −1 = γ(g(1))γ(h(1))γ(l(1))γ (g(2) · h(2)) · l(2) = RHS hence (BE5) also holds and thus Ω(A) = (H, /, .γ, fγ) is a bialgebra extending structure of A. For the ﬁnal part we use Proposition 2.7.6 as p : A n H → A, p(a n h) = aγ(h) is a bialgebra map that splits iA : A → A n H. Thus, it follows from [128, Theorem 3] that there exists an ∼ A isomorphism of bialgebras A n H = L ∗ A, where (L, *, ρL) is a bialgebra in AYD as follows: X X X L = { ai n hi ∈ A n H | ai(1) n hi(1) ⊗ ai(2) γ(hi(2) ) = ai n hi ⊗ 1A} i i i with the coalgebra structure described below:

X X ∆L( ai n hi) = (ai(1) n hi(1) ) • 1A n SA(ai(2) γ(hi(2) )) ⊗ ai(3) n hi(3) i i X X εL( ai n hi) = εA(ai)εH (hi) i i

95 CHAPTER 2. UNIFIED PRODUCTS

P A for all i ai n hi ∈ L and the structure of an object in AYD given by X X a * ai n hi = (a(1)ai n hi) • (1A n SA(a(2))) i i X X ρL( ai n hi) = ai(1) γ(hi(1) ) ⊗ ai(2) n hi(2) i i P for all a ∈ A and i ai n hi ∈ L.

Using Theorem 2.7.8 we shall construct an example of an uniﬁed product, that is also isomorphic to a biproduct, starting with a minimal set of data.

Example 2.7.9 Let G be a group, (X, 1X ) be a pointed set, such that there exists a binary operation · : X × X → X having 1X as a unit.

Let / : X × G → X be a map such that (X,/) is a right G-set with 1X / g = 1X , for all g ∈ G and γ : X → G a map with γ(1X ) = 1G such that the following compatibilities hold (x · y) · z = x / γ(y)γ(z)γ(y · z)−1 ·(y · z) (2.77) (x · y) / g = x / γ(y) g γ(y / g)−1 ·(y / g) (2.78) for all g ∈ G, x, y, z ∈ X.

Let A := k[G]. Then (H := k[X], /, .γ, fγ) is a bialgebra extending structure of k[G], where .γ, fγ are given by (2.75), (2.76), that is

−1 −1 x .γ g = γ(x)gγ(x / g) , fγ(x, y) = γ(x)γ(y)γ(x · y) for all x, y ∈ X and g ∈ G. Indeed, (2.77) and (2.78) show that the compatibility conditions (BE1) and respectively (BE3) hold. Now, the compatibilities (BE6) and (BE7) are trivially fulﬁlled as k[G] and k[X] are cocommutative coalgebras. Thus, it follows from Theorem 2.7.8 that (k[X], /, .γ, fγ) is a bialgebra extending structure of k[G] and we can construct the uniﬁed product E := k[G]nk[X] associated to (k[X], /, .γ, fγ). As a vector space E = k[G] ⊗ k[X] and has the multiplication given by (2.71) which in this case takes the form:

−1 (g n x) • (h n y) = g γ(x) h γ(y) γ (x / h) · y n(x / h) · y for all g, h ∈ G and x, y ∈ X. The coalgebra structure of E is the tensor product of the group-like coalgebras k[G] and k[X]. The multiplication on the bialgebra k[G] n k[X] can be simpliﬁed as follows: using Proposi- tion 2.7.7, we obtain that there exists an isomorphism of bialgebras

∼ k[G] n k[X] = k[G] ~ k[X]

96 2.7. UNIFIED PRODUCTS AND SPLIT EXTENSIONS OF HOPF ALGEBRAS where, the multiplication on k[G] ~ k[X] is given by (2.72): −1 (g ~ x) • (h ~ y) = gh ~ x / hγ(y) · y for all g, h ∈ G and x, y ∈ X.

Example 2.7.10 The construction of an explicit datum (G, X, ·, /, γ) as in Example 2.7.9 is a challenging problem considering the compatibilities (2.77)-(2.78) that need to be fulﬁlled by the map γ : X → G. We give such an example bellow. Let G be a group, (X, 1X ) a pointed set such that (X,/) is a right G-set with 1X / g = 1X , for all g ∈ G. Let · : X × X → X be the following binary operation:

x, if y = 1 x · y := X y, if y 6= 1X

Then, by a straightforward computation it can be seen that the compatibility conditions (2.77) and (2.78) are trivially fulﬁlled for any map γ : X → G such that γ(1X ) = 1G. Thus (H := k[X], /, .γ, fγ) is a bialgebra extending structure of k[G], where .γ, fγ are given by: −1 −1 1G, if y = 1X x .γ g = γ(x) g γ(x / g) , fγ(x, y) = γ(x)γ(y)γ(x · y) = γ(x), if y 6= 1X for all x, y ∈ X and g ∈ G.

97 CHAPTER 2. UNIFIED PRODUCTS

2.8 Coquasitriangular structures for extensions of Hopf algebras

An important class of Hopf algebras is that of quasitriangular Hopf algebras or strict quantum groups. They were introduced by Drinfeld in [66] as a remarkable tool for studying the quantum Yang-Baxter equation R12R13R23 = R23R13R12. That is, if M is a representation of a quasitriangular Hopf algebra (H,R), then the canonical map m ⊗ n 7→ P R1m ⊗ R2n is a solution for the quantum Yang-Baxter equation. The dual concept, namely that of a coquasitriangular Hopf algebra (also called braided Hopf algebras in [77], [78] or [92]) was first introduced by Majid in [102] and independently by Larson and Towber in [87]. These are Hopf algebras A endowed with a linear map p : A⊗A → k satisfying some compatibility conditions. There is, of course, a dual result concerning the quantum Yang-Baxter equation: if M is a corepresentation of a coquasitriangular Hopf algebra (A, p) then the canonical map Rp(m⊗n) = p m<1>, n<1> m<0>⊗n<0> is a solution for the quantum Yang-Baxter equation. However, what makes the coquasitriangular Hopf algebras so important is the fact that the converse of the above statement is also true. Namely, by the celebrated FRT theorem for any solution R of the quantum Yang-Baxter equation there exists a quasitriangular bialgebra (A(R), p) such that R = Rp ([81]). Based on this background, (co)quasitriangular Hopf algebras generated an explosion of interest and were studied for their implications in quantum groups, the construction of invariants of knots and 3-manifolds, statistical mechanics, quantum mechanics but they also became a subject of research in its own right. Complete descriptions of the coquasitriangular structures have been already obtained for several families of Hopf algebras, see for instance [7], [77], [78] or [92]. Among the many research topics related to coquasitriangular Hopf algebras one is of particular interest: for a given Hopf algebra H, describe (if any) all coquasitriangular structures that can be defined on H. We can formulate the more general problem: Let A ⊆ E be an extension of Hopf algebras. What is the connection between the coquasitriangular structures of A and those of E? Obviously, if (E, p) is a coquasitriangular Hopf algebra then A is also a coquasitriangular Hopf algebra with the coquasitriangular structure given by the restriction of p to A ⊗ A. The difficult part of the problem is the converse: if σ : A⊗A → k is a coquasitriangular structure on A, could it be extended to a coquasitriangular structure on E? In this section we give a complete answer to this problem in the case when the extension A ⊆ E splits: i.e. there exists π : E → A a normal left A-module coalgebra map such that π(a) = a, for all a ∈ A. We proved in Section 2.7 that an extension A ⊆ E splits in the above sense if and only if E is isomorphic to a unified product between A and a certain subcoalgebra H of E.

Let λ : H ⊗ A → k be a skew pairing between two Hopf algebras and consider Dλ(A, H) := A ./λ H to be the generalized quantum double as constructed in ([100, Example 7.2.6]). As the main application of the results in Theorem 2.8.8 the set of all coquasitriangular structures on the generalized quantum double Dλ(A, H) are completely described. In particular, it is proved that a generalized quantum double is a coquasitriangular Hopf algebra if and only if both Hopf algebras A and H are coquasitriangular. Several explicit examples are also provided. Recall from [63] that if A and H are two Hopf algebras and λ : A ⊗ H → k is a k-linear map which fulﬁlls the compatibilities:

98 2.8. COQUASITRIANGULAR STRUCTURES FOR UNIFIED PRODUCTS

(BR1) λ(xy, z) = λ(x, z(1))λ(y, z(2))

(BR2) λ(1, z) = ε(z)

(BR3) λ(x, lz) = λ(x(1), z)λ(x(2), l)

(BR4) λ(y, 1) = ε(y)

for all x, y ∈ A, l, z ∈ H, then λ is called skew pairing on (A, H) . Notice that a skew pairing λ is convolution invertible with λ−1 = λ ◦ (S ⊗ Id). Also, by a straightforward computation it can be seen that if λ is a skew pairing on (A, H) then λ ◦ (S ⊗ Id) ◦ ν is also a skew pairing on (H,A) where ν is the ﬂip map. Moreover, recall from [87] that a Hopf algebra H is called coquasitriangular or braided if there exists a linear map p : H ⊗ H → k such that relations (BR1) − (BR4) are fulﬁlled and

(BR5) p(x(1), y(1))x(2)y(2) = y(1)x(1)p(x(2), y(2))

holds for all x, y, z ∈ H. In this section we describe the coquasitriangular or braided structures on the unified product. In other words, we determine all braided structures that can be defined on the monoidal category of A n H - comodules. First we introduce some new definitions as natural generalizations for the concepts of braiding and skew pairing.

Deﬁnition 2.8.1 Let A be a Hopf algebra, H = H, ∆H , εH , 1H , · a k-module such that H, ∆H , εH is a coalgebra, H, 1H , · is an unitary not necessarily associative k-algebra, f : H ⊗ H → A a coalgebra map and p : A ⊗ A → k a braiding on A. A linear map u : A ⊗ H → k is called generalized (p,f) - right skew pairing on (A, H) if the following compatibilities are fulﬁlled for any a, b ∈ A, g, t ∈ H:

(RS1) u(ab, t) = u(a, t(1))u(b, t(2))

(RS2) u(1, h) = ε(h)

(RS3) u(a(1), g(2) · t(2))p a(2), f(g(1), t(1)) = u(a(1), t)u(a(2), g)

(RS4) u(a, 1) = ε(a)

99 CHAPTER 2. UNIFIED PRODUCTS

Deﬁnition 2.8.2 Let A be a Hopf algebra, H = H, ∆H , εH , 1H , · a k-module such that H, ∆H , εH is a coalgebra, H, 1H , · is an unitary not necessarily associative k-algebra, f : H ⊗ H → A a coalgebra map and p : A ⊗ A → k a braiding on A. A linear map v : H ⊗ A → k is called generalized (p,f) - left skew pairing on (H,A) if the following compatibilities are fulﬁlled for any b, c ∈ A, h, g ∈ H:

(LS1) p f(h(1), g(1)), c(1) v(h(2) · g(2), c(2)) = v(h, c(1))v(g, c(2))

(LS2) v(h, 1) = ε(h)

(LS3) v(h, bc) = v(h(1), c)v(h(2), b)

(LS4) v(1, a) = ε(a)

Remark 2.8.3 If H is a bialgebra and f = ε ⊗ ε is the trivial cocycle then the notion of generalized (p,f) - left/right skew pairing on (A, H) coincides with the notion of skew pairing on (A, H).

Deﬁnition 2.8.4 Let A be a Hopf algebra, H = H, ∆H , εH , 1H , · a k-module such that H, ∆H , εH is a coalgebra, H, 1H , · is an unitary not necessarily associative k-algebra, f : H ⊗ H → A a coalgebra map and p : A ⊗ A → k a braiding on A, u : A ⊗ H → k a generalized (p, f) - right skew pairing and v : H ⊗ A → k a generalized (p, f) - left skew pairing. A linear map τ : H ⊗ H → k is called a generalized (u, v) - skew braiding on H if the following compatibilities are fulﬁlled for all h, g, t ∈ H:

(SBR1) u f(h(1), g(1)), t(1) τ(h(2) · g(2), t(2)) = τ(h, t(1))τ(g, t(2))

(SBR2) τ(1, g) = ε(g)

(SBR3) τ(h(1), g(2) · t(2))v h(2), f(g(1), t(1)) = τ(h(1), t)τ(h(2), g)

(SBR4) τ(g, 1) = ε(g)

(SBR5) τ(h(1), g(1))h(2) · g(2) = g(1) · h(1)τ(h(2), g(2))

Remark 2.8.5 If H is a bialgebra and f = ε ⊗ ε is the trivial cocycle then the notion of generalized (u, v) - skew braiding on H coincides with the notion of coquasitriangular structure (or braiding) on H.

100 2.8. COQUASITRIANGULAR STRUCTURES FOR UNIFIED PRODUCTS

Theorem 2.8.6 Let A be a Hopf algebra and Ω(A) = (H, /, ., f) a Hopf algebra extending structure of A. There is a bijective correspondence between: (i) The set of all coquasitriangular structures σ on the uniﬁed product A n H; (ii) The set of all quadruples (p, τ, u, v) where p : A⊗A → k, τ : H ⊗H → k, u : A⊗H → k, v : H ⊗ A → k are linear maps such that (A, p) is a coquasitriangular Hopf algebra, u is a generalized (p, f) - right skew pairing, v is a generalized (p, f) - left skew pairing, (H, τ) is a generalized (u, v) - skew braiding and the following compatibilities are fulﬁlled:

v(h(1), b(1))(h(2) . b(2)) ⊗ (h(3) / b(3)) = b(1) ⊗ h(1)v(h(2), b(2)) (2.79)

(g(1) . a(1)) ⊗ (g(2) / a(2))u(a(3), g(3)) = u(a(1), g(1))a(2) ⊗ g(2) (2.80)

τ(h(1), g(1))f(h(2), g(2)) = f(g(1), h(1))τ(h(2), g(2)) (2.81)

u(a(1), g(2) / c(2))p(a(2), g(1) . c(1)) = p(a(1), c)u(a(2), g) (2.82)

τ(h(1), g(2) / c(2))v(h(2), g(1) . c(1)) = v(h(1), c)τ(h(2), g) (2.83)

p(h(1) . b(1), c(1))v(h(2) / b(2), c(2)) = v(h, c(1))p(b, c(2)) (2.84)

u(h(1) . b(1), t(1))τ(h(2) / b(2), t(2)) = τ(h, t(1))u(b, t(2)) (2.85)

Under the above bijection the coquasitriangular structure σ :(A n H) ⊗ (A n H) → k corresponding to (p, τ, u, v) is given by:

σ(a n h, b n g) = u(a(1), g(1))p(a(2), b(1))τ(h(1), g(2))v(h(2), b(2)) (2.86) for all a, b, c ∈ A and h, g, t ∈ H.

Proof: Suppose ﬁrst that (A n H, σ) is a coquasitriangular Hopf algebra. We deﬁne the following linear maps:

p : A ⊗ A → k, p(a, b) = σ(a ⊗ 1, b ⊗ 1) τ : H ⊗ H → k, τ(h, g) = σ(1 ⊗ h, 1 ⊗ g) u : A ⊗ H → k, u(a, h) = σ(a ⊗ 1, 1 ⊗ h) v : H ⊗ A → k, v(h, a) = σ(1 ⊗ h, a ⊗ 1)

Before going into the proof we collect here some compatibilities satisfied by the maps defined above which will be useful in the sequel. The following are just easy consequences of the fact that σ is a coquasitriangular structure on A n H and, hence, it satisfies the normalizing relations (BR2) and (BR4):

p(1, b) = ε(b) = p(b, 1) (2.87) τ(1, h) = ε(h) = τ(h, 1) (2.88) u(1, h) = ε(h), u(a, 1) = ε(a) (2.89) v(1, a) = ε(a), v(h, 1) = ε(h) (2.90)

Remark that from relation (2.88) it follows that τ fulﬁlls (SBR2) and (SBR4) while from relation (2.90) we can derive that v fulﬁlls (LS2) and (LS4).

101 CHAPTER 2. UNIFIED PRODUCTS

First we prove that relation (2.86) indeed holds:

σ(a#h, b#g) = σ(a#1)(1#h), (b#1)(1#g)= (BR1) = σ a#1, (b(1)#1)(1#g(1)) σ (1#h), (b(2)#1)(1#g(2)) (BR3) = σ(a(1)#1, 1#g(1))σ(a(2)#1, b(1)#1)σ(1#h(1), 1#g(2))σ 1#h(2), b(2)#1)

= u(a(1), g(1))p(a(2), b(1))τ(h(1), g(2))v(h(2), b(2)) Next we prove that (A, p) is a coquasitriangular Hopf algebra, u is a generalized (p, f) - right skew pairing on (H,A), v is a generalized (p, f) - left skew pairing on (A, H) and τ is a generalized (u, v) - skew braiding on H. Having in mind that (A n H, σ) is a coquasitriangular Hopf algebra it is straightforward to see that (A, p) is a coquasitriangular Hopf algebra by considering x = a#1, y = b#1 and z = c#1 in (BR1) − (BR5). Since σ satisﬁes (BR1), then for all a, b, c ∈ A and h, g, t ∈ H we have: σ a(g(1) . b(1))f(h(2) / b(2), h(1))#(g(3) / b(3)) · h(2), c#t =

= σ(a ⊗ g, c(1) ⊗ t(1))σ(b ⊗ h, c(2) ⊗ t(2)) (2.91) Moreover, since σ also fulﬁlls (BR3) we have: σ a#h, b(g(1) . c(1))f(g(2) / c(2), t(1))#(g(3) / c(3)) · t(2) =

= σ(a(1) ⊗ h(1), c ⊗ t)σ(a(2) ⊗ h(2), b ⊗ g) (2.92) Furthermore, by (BR5) we have:

σ(a(1) ⊗ h(1), b(1) ⊗ g(1))a(2)(h(2) . b(2))f(h(3) / b(3), g(2)) ⊗ (h(4) / b(4)) · g(3)

= b(1)(g(1) . a(1))f(g(2) / a(2), h(1)) ⊗ (g(3) / a(3)) · h(2)σ(a(4) ⊗ h(3), b(2) ⊗ g(4)) (2.93) By considering h = g = 1 and c = 1 in (2.91) we get relation (RS1). If we let b = c = 1 and h = 1 in (2.92) yields: u(a(1), g(3) · t(3))p a(2), f(g(1), t(1)) τ(1, g(4)t(4))v 1, f(g(2), t(2)) = u(a(1), t)u(a(2), g) Now using relations (2.88) and (2.90) we get (RS3). Hence we proved that u is a generalized (p, f) - right skew pairing on (H,A). Considering a = b = 1 and t = 1 in (2.91) yields: u f(h(1), g(1)), 1 p f(h(2), g(2)), c(1) τ(h(3) · g(3), 1)v(h(4) · g(4), c(2)) = v(h, c(1))v(g, c(2)) Using (2.88) and (2.89) we get that (LS1) holds for v. Moreover from (2.92) applied to g = t = 1 and a = 1 we get that (LS3) also holds for v and we proved that v is indeed a generalized (p, f) - left skew pairing on (A, H). Next we apply (2.91) for a = b = c = 1: u f(h(1), g(1)), t(1) p f(h(2), g(2)), 1 τ(h(3) · g(3), t(2))v(h(4) · g(4), 1) = τ(h, t(1))τ(g, t(2))

102 2.8. COQUASITRIANGULAR STRUCTURES FOR UNIFIED PRODUCTS

Using (2.87) and (2.90) we obtain (SBR1). Now (2.92) applied for a = b = c = 1 yields:

u(1, g(3) · t(3))p 1, f(g(1), t(1)) v h(2), f(g(2), t(2)) τ(h(1), g(4) · t(4)) = τ(h(1), t)τ(h(2), g)

From (2.87) and (2.89) we obtain that (SBR3) holds for τ. Considering a = b = 1 in (2.93) we get:

τ(h(1), g(1))f(h(2), g(2))#h(3) · g(3) = f(g(1), h(1))#g(2) · h(2)τ(h(3), g(3))

Having in mind that f is a coalgebra map we obtain, by applying ε ⊗ Id, that (SBR5) holds for τ and therefore τ is a generalized (u, v) - skew braiding. We still need to prove that the compatibilities (2.79)-(2.85) hold. Compatibilities (2.79)-(2.80) are obtained from (2.93) by considering: a = 1 and g = 1 respectively b = 1 and h = 1 while (2.81) can be derived from (2.93) by considering a = b = 1 and then applying Id ⊗ ε. The next two compatibilities, (2.82) and (2.83), can be obtained by considering h = t = 1 and b = 1 respectively a = b = 1 and t = 1 in (2.92). To this end, relations (2.84) and (2.85) can be derived from (2.91) by considering g = t = 1 and a = 1 respectively a = c = 1 and g = 1. Assume now that (A, p) is a coquasitriangular Hopf algebra, u is a generalized (p, f) - right skew pairing, v is a generalized (p, f) - left skew pairing, τ is a generalized (u, v) - skew braiding and σ is given by (2.86) such that compatibilities (2.79)-(2.85) are fulﬁlled. Then, using relations (RS2), (SBR2), (LS2) and the fact that p is a coquasitriangular structure we can prove that for all a ∈ A, h ∈ H we have:

σ(1#1, a#h) = u(1, h(1))p(1, a(1))τ(1, h(2))v(1, a(2)) = ε(a)ε(h) = ε(a#h)

Moreover, using relations (RS4), (SBR4), (LS4) and again the fact that p is a coquasitriangular structure, we also have:

σ(a#h, 1#1) = u(a(1), 1)p(a(2), 1)τ(h(1), 1)v(h(2), 1) = ε(a)ε(h) = ε(a#h) for all a ∈ A, h ∈ H. Hence σ also fulfills (BR4). To prove that σ satisfies (BR1) we start by first computing the left hand side. Thus for all a, b, c ∈ A and h, g, t ∈ H we have:

103 CHAPTER 2. UNIFIED PRODUCTS

LHS = u a(1)(g(1) . b(1))f(g(3) / b(3), h(1)), t(1) v (g(6) / b(6)) · h(4), c(2) p a(2)(g(2) . b(2))f(g(4) / b(4), h(2)), c(1) τ (g(5) / b(5)) · h(3), t(2) (RS1) = u(a(1), t(1))u(g(1) . b(1), t(2))u f(g(3) / b(3), h(1)), t(3) v (g(6) / b(6)) · h(4), c(4) p(a(2), c(1))p f(g(4) / b(4), h(2)), c(3) τ (g(5) / b(5)) · h(3), t(4) p(g(2) . b(2), c(2)) (BE7) = u(a(1), t(1))u(g(1) . b(1), t(2))u f(g(3) / b(3), h(1)), t(3) v (g(6) / b(6)) · h(4), c(4) p(a(2), c(1))p f(g(5) / b(5), h(3)), c(3) τ (g(4) / b(4)) · h(2), t(4) p(g(2) . b(2), c(2)) (LS1) = u(a(1), t(1))u(g(1) . b(1), t(2))u f(g(3) / b(3), h(1)), t(3) p(a(2), c(1)) p(g(2) . b(2), c(2))τ (g(4) / b(4)) · h(2), t(4) v(g(5) / b(5), c(3))v(h(3), c(4)) (SBR1) = u(a(1), t(1))u(g(1) . b(1), t(2))p(a(2), c(1))p(g(2) . b(2), c(2))

τ(g(3) / b(3), t(3))τ(h(1), t(4))v(g(4) / b(4), c(3))v(h(2), c(4)) (BE6) = u(a(1), t(1))u(g(1) . b(1), t(2))p(a(2), c(1))p(g(3) . b(3), c(2))

τ(g(2) / b(2), t(3))τ(h(1), t(4))v(g(4) / b(4), c(3))v(h(2), c(4)) (2.84) = u(a(1), t(1))u(g(1) . b(1), t(2))p(a(2), c(1))τ(g(2) / b(2), t(3))τ(h(1), t(4))

v(g(3), c(2))p(b(3), c(3))v(h(2), c(4)) (2.85) = u(a(1), t(1))p(a(2), c(1))τ(g(1), t(2))u(b(1), t(3))τ(h(1), t(4))

v(g(2), c(2))p(b(2), c(3))v(h(2), c(4)) = RHS where in the second equality we also used the fact that p is a coquasitriangular structure. To prove (BR3) we start again by computing the left hand side. Thus for all a, b, c ∈ A and h, g, t ∈ H we have:

LHS = u (a(1), (g(5) / c(5)) · t(3) p a(2), b(1)(g(1) . c(1))f(g(3) / c(3), t(1)) τ h(1), (g(6) / c(6)) · t(4) v h(2), b(2)(g(2) . c(2))f(g(4) / c(4), t(2)) (RS3) = u a(1), (g(5) / c(5)) · t(3) p a(2), f(g(3) / c(3), t(1)) p(a(3), g(1) . c(1))p(a(4), b(1)) τ h(1), (g(6) / c(6)) · t(4) v h(2), f(g(4) / c(4), t(2)) v(h(3), g(2) . c(2))v(h(4), b(2)) (BE7) = u a(1), (g(4) / c(4)) · t(2) p a(2), f(g(3) / c(3), t(1)) p(a(3), g(1) . c(1))p(a(4), b(1)) τ h(1), (g(6) / c(6)) · t(4) v h(2), f(g(5) / c(5), t(3)) v(h(3), g(2) . c(2))v(h(4), b(2))

104 2.8. COQUASITRIANGULAR STRUCTURES FOR UNIFIED PRODUCTS

(SBR3) = u a(1), (g(4) / c(4)) · t(2) p a(2), f(g(3) / c(3), t(1)) p(a(3), g(1) . c(1))p(a(4), b(1))

τ(h(1), t(3))τ(h(2), g(5) / c(5))v(h(3), g(2) . c(2))v(h(4), b(2)) (RS3) = u(a(1), t(1))u(a(2), g(3) / c(3))p(a(3), g(1) . c(1))p(a(4), b(1))τ(h(1), t(2))

τ(h(2), g(4) / c(4))v(h(3), g(2) . c(2))v(h(4), b(2)) (BE6) = u(a(1), t(1))u(a(2), g(2) / c(2))p(a(3), g(1) . c(1))p(a(4), b(1))τ(h(1), t(2))

τ(h(2), g(4) / c(4))v(h(3), g(3) . c(3))v(h(4), b(2)) (2.82) = u(a(1), t(1))p(a(2), c(1))u(a(3), g(1))p(a(4), b(1))τ(h(1), t(2))

τ(h(2), g(3) / c(3))v(h(3), g(2) . c(2))v(h(4), b(2)) (2.83) = u(a(1), t(1))p(a(2), c(1))u(a(3), g(1))p(a(4), b(1))τ(h(1), t(2))

v(h(2), c(2))τ(h(3), g(2))v(h(4), b(2)) = RHS

Note that in the second equality we used the fact that p is a coquasitriangular structure. In order to show that σ also fulﬁlls (BR5) we need the following compatibilities that can be easily derived from (2.79) and (2.80) by applying ε ⊗ Id:

v(h(1), b(1))(h(2) / b(2)) = h(1)v(h(2), b) (2.94)

(g(1) / a(1))u(a(2), g(2)) = u(a, g(1))g(2) (2.95)

Computing the left hand side of (BR5) we obtain:

LHS = u(a(1), g(1))p(a(2), b(1))τ(h(1), g(2))v(h(2), b(2))a(3)(h(3) . b(3))

f(h(4) / b(4), g(3))#(h(5) / b(5)) · g(4) (BE7) = u(a(1), g(1))p(a(2), b(1))τ(h(1), g(2))v(h(2), b(2))a(3)(h(3) . b(3))

f(h(5) / b(5), g(4))#(h(4) / b(4)) · g(3) (2.79) = u(a(1), g(1))p(a(2), b(1))τ(h(1), g(2))a(3)b(2)f(h(4) / b(4), g(4))#

h(2) · g(3)v(h(3), b(3))

= u(a(1), g(1))τ(h(1), g(2))b(1)a(2)p(a(3), b(2))f(h(4) / b(4), g(4))#

h(2) · g(3)v(h(3), b(3)) (SBR5) = u(a(1), g(1))b(1)a(2)p(a(3), b(2))f(h(4) / b(4), g(4))#(g(2) · h(1))

τ(h(2), g(3))v(h(3), b(3))

105 CHAPTER 2. UNIFIED PRODUCTS

(2.80) = b(1)(g(1) . a(1))p(a(4), b(2))f(h(4) / b(4), g(5))#(g(2) / a(2)) · h(1)

u(a(3), g(3))τ(h(2), g(4))v(h(3), b(3)) (2.95) = b(1)(g(1) . a(1))p(a(3), b(2))f(h(4) / b(4), g(5))#(g(3) · h(1))u(a(2), g(2))

τ(h(2), g(4))v(h(3), b(3)) (2.94) = b(1)(g(1) . a(1))p(a(3), b(2))f(h(3), g(5))#(g(3) · h(1))u(a(2), g(2))

τ(h(2), g(4))v(h(4), b(3)) (2.81) = b(1)(g(1) . a(1))p(a(3), b(2))v(h(4), b(3))τ(h(3), g(5))f(g(4), h(2))#

(g(3) · h(1))u(a(2), g(2)) (BE7) = b(1)(g(1) . a(1))p(a(3), b(2))v(h(4), b(3))τ(h(3), g(5))f(g(3), h(1))#

(g(4) · h(2))u(a(2), g(2)) (2.95) = b(1)(g(1) . a(1))p(a(4), b(2))v(h(4), b(3))τ(h(3), g(5))f(g(2) / a(2), h(1))#

(g(4) · h(2))u(a(3), g(3)) (2.95) = b(1)(g(1) . a(1))p(a(5), b(2))v(h(4), b(3))τ(h(3), g(5))f(g(2) / a(2), h(1))# (g(3) / a(3)) · h(2) u(a(4), g(4) = RHS

In the forth equality we used the fact that p is a coquasitriangular structure. Thus (BR5) holds for σ and this ends the proof.

The following result which characterizes the coquasitriangular structures on a bicrossed product of Hopf algebras can be obtained from Theorem 2.8.6 by considering f = ε ⊗ ε to be the trivial cocycle.

Corollary 2.8.7 Let A ./ H be a bicrossed product of Hopf algebras. There is a bijective correspondence between: (i) The set of all coquasitriangular structures σ on the bicrossed product A ./ H; (ii) The set of all quadruples (p, τ, u, v), where p : A⊗A → k, τ : H ⊗H → k, u : A⊗H → k, v : H⊗A → k are linear maps such that (A, p) and (H, τ) are coquasitriangular Hopf algebras, u and v are skew pairings on (A, H) respectively on (H,A) and the following compatibilities are fulﬁlled:

v(h(1), b(1))(h(2) . b(2)) ⊗ (h(3) / b(3)) = b(1) ⊗ h(1)v(h(2), b(2))

(g(1) . a(1)) ⊗ (g(2) / a(2))u(a(3), g(3)) = u(a(1), g(1))a(2) ⊗ g(2)

u(a(1), g(2) / c(2))p(a(2), g(1) . c(1)) = p(a(1), c)u(a(2), g)

106 2.8. COQUASITRIANGULAR STRUCTURES FOR UNIFIED PRODUCTS

τ(h(1), g(2) / c(2))v(h(2), g(1) . c(1)) = v(h(1), c)τ(h(2), g)

p(h(1) . b(1), c(1))v(h(2) / b(2), c(2)) = v(h, c(1))p(b, c(2))

u(h(1) . b(1), t(1))τ(h(2) / b(2), t(2)) = τ(h, t(1))u(b, t(2))

Under the above bijection the coquasitriangular structure σ :(A ./ H) ⊗ (A ./ H) → k corresponding to (p, τ, u, v) is given by:

σ(a n h, b n g) = u(a(1), g(1))p(a(2), b(1))τ(h(1), g(2))v(h(2), b(2)) (2.96) for all a, b, c ∈ A and h, g, t ∈ H.

107 CHAPTER 2. UNIFIED PRODUCTS

Applications: coquasitriangular structures on generalized quantum doubles

Let A and H be two Hopf algebras and λ : H ⊗ A → k be a skew pairing. Then (A, H) is a matched pair of Hopf algebras with the following two actions:

−1 h / a = h(2)λ (h(1), a(1))λ(h(3), a(2)) −1 h . a = a(2)λ (h(1), a(1))λ(h(2), a(3)) The corresponding double cross product is called the generalized quantum double and it will be denoted by A ./λ H ([100, Example 7.2.6]). As a special case of Corollary 2.8.7 we get:

Theorem 2.8.8 Let A and H be two Hopf algebras and λ : H ⊗ A → k be a skew pairing. There is a bijective correspondence between:

(i) The set of all coquasitriangular structures σ on the generalized quantum double A ./λ H; (ii) The set of all quadruples (p, τ, u, v), where p : A⊗A → k, τ : H ⊗H → k, u : A⊗H → k, v : H⊗A → k are linear maps such that (A, p) and (H, τ) are coquasitriangular Hopf algebras, u and v are skew pairings on (A, H) respectively on (H,A) and the following compatibilities are fulﬁlled:

−1 v(h(1), b(1))b(3) ⊗ λ (h(2), b(2))λ(h(4), b(4))h(3) = b(1) ⊗ h(1)v(h(2), b(2)) (2.97) −1 a(2)λ (g(1), a(1))λ(g(3), a(3)) ⊗ g(2)u(a(4), g(4)) = u(a(1), g(1))a(2) ⊗ g(2) (2.98) −1 u(a(1), g(2))λ(g(3), c(3))p(a(2), c(2))λ (g(1), c(1)) = p(a(1), c)u(a(2), g) (2.99) −1 τ(h(1), g(2))λ(g(3), c(3))v(h(2), c(2))λ (g(1), c(1)) = v(h(1), c)τ(h(2), g) (2.100) −1 p(b(2), c(1))λ (h(1), b(1))v(h(2), c(2))λ(h(3), b(3)) = p(b, c(2))λ(h, c(1)) (2.101) −1 u(b(2), t(1))λ (h(1), b(1))τ(h(2), t(2))λ(h(3), b(3)) = τ(h, t(1))u(b, t(2)) (2.102)

Under this correspondence the coquasitriangular structure σ :(A ./λ H) ⊗ (A ./λ H) → k corresponding to (p, τ, u, v) is given by:

σ(a ⊗ h, b ⊗ g) = u(a(1), g(1))p(a(2), b(1))τ(h(1), g(2))v(h(2), b(2)) (2.103) for all a, b, c ∈ A and h, g, t ∈ H.

Theorem 2.8.9 Let (A, p) and (H, τ) be two coquasitriangular Hopf algebras and λ : H⊗A → k be a skew pairing. Then the generalized quantum double A ./λ H is a coquasitriangular Hopf algebra with the coquasitriangular structure given by: σ(a ./ h, b ./ g) = λ S(g(1), a(1)) p(a(2), b(1))τ(h(1), g(2))λ(h(2), b(2)) (2.104)

Proof: We make use of Theorem 2.8.8: take v := λ and u := λ−1 ◦ ν, where ν is the ﬂip map. We need to prove that relations (2.97)-(2.102) are fulﬁlled. We have: LHS(2.97) = b(3) ⊗ λ(h(1), b(1))λ S(h(2)), b(2) λ(h(4), b(4))h(3)

= b(1) ⊗ h(1)λ(h(2), b(2)) = RHS(2.97)

108 2.8. APPLICATIONS: THE GENERALIZED QUANTUM DOUBLES

LHS(2.98) = a(2)λ S(g(1)), a(1) λ(g(3), a(3))λ S(g(4)), a(4) ⊗ g(2) = a(2)λ S(g(1)), a(1) ⊗g(2) = RHS(2.98)

LHS(2.99) = λ S(g(2)), a(1) λ(g(3), c(3))p(a(2), c(2))λ S(g(1)), c(1) = λ S(g(1)), c(1)a(1) p(a(2), c(2))λ(g(2), c(3)) = λ S(g(1)), a(2)c(2) p(a(1), c(1))λ(g(2), c(3)) = λ S(g(2)), c(2) λ S(g(1)), a(2) p(a(1), c(1))λ(g(3), c(3)) = λ S(g(2))g(3), c(2) λ S(g(1)), a(2) p(a(1), c(1)) = λ S(g), a(2) p(a(1), c) = RHS(2.99)

LHS(2.100) = τ(h(1), g(2))λ(g(3), c(3))λ(h(2), c(2))λ S(g(1)), c(1) = τ(h(1), g(2))λ(h(2)g(3), c(2))λ S(g(1)), c(1) = τ(h(2), g(3))λ(g(2)h(1), c(2))λ S(g(1)), c(1) = τ(h(2), g(3))λ(g(2), c(2))λ(h(1), c(3))λ S(g(1)), c(1)

= τ(h(2), g)λ(h(1), c) = RHS(2.101)

LHS(2.101) = p(b(2), c(1))λ S(h(1)), b(1) λ(h(2), c(2))λ(h(3), b(3)) = p(b(2), c(1))λ S(h(1)), b(1) λ(h(2), b(3)c(2)) = p(b(3), c(2))λ S(h(1)), b(1) λ(h(2), c(1)b(2)) = p(b(3), c(2))λ S(h(1)), b(1) λ(h(2), b(2))λ(h(3), c(1))

= p(b, c(2))λ(h, c(1)) = RHS(2.101)

LHS(2.102) = λ S(t(1)), b(2) λ S(h(1)), b(1) τ(h(2), t(2))λ(h(3), b(3)) = λ S(t(1)h(1)), b(1) τ(h(2), t(2))λ(h(3), b(2))

= λ S(t(2))S(h(2)), b(1) τ(h(1), t(1))λ(h(3), b(2)) = λ S(t(2)), b(1) λ(S(h(2)), b(2))τ(h(1), t(1))λ(h(3), b(3)) = τ(h, t(1))λ S(t(2)), b = RHS(2.102)

As a consequence, we derive the necessary and sufﬁcient conditions for the generalized quantum double to be a coquasitriangular Hopf algebra.

109 CHAPTER 2. UNIFIED PRODUCTS

Corollary 2.8.10 Let A and H be two Hopf algebras and τ : H ⊗ A → k be a skew pairing. Then the generalized quantum double A ./τ H is a coquasitriangular Hopf algebra if and only if both Hopf algebras A and H are coquasitriangular.

Also as a special case of Theorem 2.8.9 we recover Majid’s result [100, Proposition 7.3.1]:

Corollary 2.8.11 Let (A, p) be a coquasitriangular Hopf algebra. Then the generalized quantum double A ./p A has a coquasitriangular structure given by: σ(a ⊗ b, c ⊗ d) = p S(d(1)), a(1) p(a(2), c(1))p(b(1), d(2))p(b(2), c(2))

Proof: Consider A = H and σ = τ := p in Theorem 2.8.9.

Examples 2.8.12 1) Consider the group algebra kZ with the obvious Hopf algebra structure and let g be a generator of Z in multiplicative notation. We have a coquasitriangular structure t l tl p : kZ ⊗ kZ → k given by: p(g , g ) = q . Now consider the polynomial algebra k[X] with the coalgebra structure and the antipode given by: n X n ∆(Xn) = Xk ⊗ Xn−k, ε(Xn) = 0,S(Xn) = (−1)nXn, for all n > 0 k k=0 Any element α ∈ k, induces a coquasitriangular structure τ on k[X] as follows: 0, if i 6= j τ(Xi,Xj) = i!αi, if i = j

Moreover, there is a skew pairing λ between the two Hopf algebras k[X] and kZ given by: λ(Xm, gt) = tm with the convention that t0 = 1 even if t = 0. Thus, by applying Theorem 2.8.9 we obtain a coquasitriangular structure σ on the generalized quantum double kZ ./λ k[X]: r+k=m X mn σ(gt ⊗ Xn, gl ⊗ Xm) = (−1)k tkqtlln−rr!αr k r k∈0,m,r∈0,n

2) Let k be a ﬁeld with chark 6= 2 and H4 be Sweedler’s Hopf algebra. That is, H4 is generated as an algebra by elements g and x subject to relations: g2 = 1, x2 = 0, xg = −gx The coalgebra structure and the antipode are given by: ∆(g) = g ⊗ g, ∆(x) = x ⊗ g + 1 ⊗ x, ε(g) = 1, ε(x) = 0 S(g) = g, S(x) = gx

For any α ∈ k the map pα : H4 ⊗ H4 → k is a coquasitriangular structure on H4, where pα is deﬁned as:

110 2.8. APPLICATIONS: THE GENERALIZED QUANTUM DOUBLES

pα 1 g x gx 1 1 1 0 0 g 1 -1 0 0 x 0 0 α α gx 0 0 α α

Let α, β, γ ∈ k and consider pα, pβ, pγ the corresponding coquasitriangular structures on H4. Since any coquasitriangular structure is in particular a skew pairing, we can construct the generalized quantum double H4 ./pγ H4. In view of Theorem 2.8.9 there is a coquasitriangular structure on H4 ./pγ H4 given by: σ(a ⊗ h, b ⊗ g) = pγ S(g(1)), a(1) pα(a(2), b(1))pβ(h(1), g(2))pγ(h(2), b(2))

3) Let k be a ﬁeld with char(k) 6= 2 and consider the k-algebra U](n) deﬁned by generators {c, x1, ..., xn, y1, ..., yn} and relations: 2 2 2 c = 1, xi = yi = 0, cxi + xic = 0, cyi + yic = 0 xixj + xjxi = 0, yiyj + yjyi = 0, xiyj = yjxi, 1 ≤ i ≤ n

U](n) has a Hopf algebra structure given by:

∆(c) = c ⊗ c, ∆(xi) = 1 ⊗ xi + xi ⊗ c, ∆(yi) = c ⊗ yi + yi ⊗ 1

ε(c) = 1, ε(xi) = ε(yi) = 0,S(c) = c, S(xi) = cxi,S(yi) = yic, 1 ≤ i ≤ n

U](n) is a quotient of the Hopf algebra U(n) introduced by Takeuchi in [144]. Now consider Bf− and Bf+ to be the Hopf subalgebras of U](n) generated by {c, y1, ..., yn} respectively {c, x1, ..., xn}. These are the so-called Borel subalgebras . Bf− and Bf+ are coquasitriangular Hopf algebras with:

τ : Bf−⊗Bf− → k, τ(c, c) = −1, τ(c, yi) = τ(yj, c) = 0, τ(yi, yj) = αij, 1 ≤ i, j ≤ n p : Bf+⊗Bf+ → k, p(c, c) = −1, p(c, xi) = p(xj, c) = 0, p(xi, xj) = βij, 1 ≤ i, j ≤ n

Moreover, there is a skew-pairing λ : Bf− ⊗ Bf+ → k given by:

λ(c, c) = −1, λ(c, xj) = λ(yi, c) = 0, λ(yi, xj) = δij, 1 ≤ i, j ≤ n where δi,j is the Kronecker delta. Therefore, using Theorem 2.8.9 the generalized quantum double Bf+ ./τ Bf− is a coquasitriangular Hopf algebra with the coquasitriangular structure σ :(Bf+ ./τ Bg−) ⊗ (Bf+ ./τ Bf−) → k given by:

σ c ./ c c ./ yi xj ./ c xk ./ yl c ./ c 1 0 0 0 c ./ ys 0 αsi δsj 0 xm ./ c 0 −δmj βmj 0 xn ./ yr 0 0 0 αrlβnk − δrkδln

111 CHAPTER 2. UNIFIED PRODUCTS

2.9 Crossed product of Hopf algebras

The crossed product is a fundamental construction in mathematics. It was first introduced in group theory related to the famous extension problem of Holder: any extension (E, i, π) of a α group H by a group G is equivalent to a crossed product extension (H#f G, iH , πG). The construction of crossed products of groups has served as a model for later generalizations at the level of groups acting on rings [122], Hopf algebras acting on k-algebras [35], von Neumann algebras [107], etc. . The crossed product A#f H of a Hopf algebra H acting on a k-algebra A was introduced independently in [35] and [62] as a generalization of the crossed product of groups acting on k-algebras. It has only an algebra structure and was studied mainly as an algebra extension of A, being an essential tool in Hopf-Galois extensions theory as it is well known that Hopf-Galois extensions with normal basis are equivalent to crossed products with invertible cocycle. Many algebraic properties of the crossed product of a Hopf algebra H acting on a k-algebra A such as semisimplicity, semiprimeness, etc. were studied in this setting ([52], [127]). If, in addition, A and H are Hopf algebras and the cocycle f : H ⊗ H → A and the action . : H ⊗ A → A are coalgebra maps satisfying two compatibility conditions then we proved in Example 2.5.6, 2) . that the crossed product A#f H has a natural Hopf algebra structure which we call the crossed product of Hopf algebras. An important feature of the crossed product of Hopf algebras is the following: a Hopf algebra E is isomorphic as a Hopf algebra to a crossed product of Hopf algebras if and only if E factorizes through a normal Hopf subalgebra and a subcoalgebra containing the unit of E (Theorem 2.9.3). . It turns out that the crossed product of Hopf algebras A#f H is also a special case of the so called ”cocycle bicrossproduct bialgebra” constructed for bialgebras by Majid and Soibelman in [99, Theorem 2.9] if we let the cocylce cross coproduct to be the trivial one, i.e. the coalgebra structure is the tensor product of coalgebras, and the action . and the cocycle f to be coalgebra maps. A remarkable example is the quantum Weyl algebra which was shown to have a cocycle bicrossproduct bialgebra structure in [99, Corollary 3.4]. Later on, an antipode for the cocycle bicrossproduct bialgebra was constructed by Andruskiewitsch and Devoto in [21] where they study short exact sequences for quantum groups. For a further use recall that a Hopf algebra H is called (co)semisimple if the underlying (co)algebra structure is (co)semisimple. An element t in a finite-dimensional Hopf algebra H is called right integral in H if tx = ε(x)t for all x ∈ H. Maschke’s theorem states that a finite-dimensional Hopf algebra H is semisimple if and only if ε(t) 6= 0, for some right integral t ∈ H.

Crossed product of Hopf algebras

Let H be a Hopf algebra, A a k-algebra and . : H ⊗ A → A, f : H ⊗ H → A two k- linear maps such that the normalizing conditions (2.24), (2.25) are fulﬁlled as well as the twisted module condition (2.27) and the cocycle condition (2.28). We already saw in Section 2.4 that the crossed product of a Hopf algebra H acting on a k-algebra A that A#f H is an associative algebra with identity element 1A#1H .

112 2.9. CROSSED PRODUCT OF HOPF ALGEBRAS

If, in addition A is also a Hopf algebra and the maps . : H ⊗ A → A and f : H ⊗ H → A . are coalgebra morphisms, then the crossed product A#f H has a Hopf algebra structure with the coalgebra structure given by the tensor product of coalgebras if and only if the following two compatibility conditions hold:

g(1) ⊗ g(2) . a = g(2) ⊗ g(1) . a (2.105)

g(1)h(1) ⊗ f(g(2), h(2)) = g(2)h(2) ⊗ f(g(1), h(1)) (2.106) for all g, h ∈ H and a, b ∈ A. Remark that both relations (2.105) and (2.106) are trivially fulﬁlled if H is cocommutative. As a special case by considering / to be the trivial action in . Proposition 2.5.9 we obtain that the antipode of the Hopf algebra A#f H is given by:

S(a#g) := SA f SH (g(2)), g(3) #SH (g(1)) · SA(a)#1H (2.107) for all a ∈ A and g ∈ H.

Deﬁnition 2.9.1 A quadruple (A, H, ., f), where A and H are Hopf algebras and . : H ⊗A → A, f : H ⊗ H → A are two coalgebra maps such that relations (2.24)-(2.25), (2.27)-(2.28) and (2.105)-(2.106) are fulﬁlled is called crossed system of Hopf algebras . The corresponding crossed product of Hopf algebras associated to the crossed system (A, H, ., f) will be denoted . by A#f H.

Examples 2.9.2 1) Let A, H be two Hopf algebras and ., f be the trivial action respectively the trivial cocycle , that is: a . h = ε(a)h and f(g, h) = ε(g)ε(h) for all a ∈ A and g, h ∈ H. Then the associated crossed product is exactly the tensor product of Hopf algebras A ⊗ H. 2) Let A, H be two Hopf algebras and f be the trivial action. If A is an H-module algebra via the coalgebra map . and the relation (2.106) is fulfilled then the crossed product has the algebra structure given by the smash product ([105]) and it will be denoted by A#.H. 3) First recall that for any set X, the free module of basis X, kX can be made into a coalgebra by considering the structure given by: ∆(x) = x ⊗ x and ε(x) = 1, for all x ∈ X. This will . be the coalgebra structure we consider on kG, kH as well as on k[H#f G]. In this setting, if (H, G, f, .) is a normalized crossed system of groups (see [16] for definitions and further properties) then (kH, kG, f,e e.) is a crossed system of Hopf algebras where feand e. are obtained by linearizing the maps f and .. Furthermore, there exists an isomorphism of Hopf algebras: ϕ : k[H ×. G] → kH#e. kG given by ϕ(h, g)= h#g for all h ∈ H, g ∈ G and extended f fe . linearly to k[H ×f G]. Moreover, since coalgebra maps between two group algebras are induced by maps between the corresponding groups of the group algebras then any crossed product of Hopf algebras between group algebras arises as above. 4) Let A and H be two Hopf algebras such that H is cocommutative and γ : H → A a unitary coalgebra map. Define:

−1 . := .γ : H ⊗ A → A, h . a := γ(h(1))aγ (h(2))

113 CHAPTER 2. UNIFIED PRODUCTS

−1 f := fγ : H ⊗ H → A, f(h, g) = γ(h(1))γ(g(1))γ (h(2)g(2))

Then, using the cocommutativity of H, is a routine computation to check that (A, H, .γ, fγ) is a crossed system of Hopf algebras and, moreover, the map given by:

. ϕ : A#f H → A ⊗ H, ϕ(a#h) = aγ(h(1)) ⊗ h(2) for all a ∈ A and h ∈ H is an isomorphism of Hopf algebras.

Basic properties of crossed products

In this section we collect some fundamental properties of crossed products of Hopf algebras. For . a crossed product A#f H we can deﬁne the following k-linear maps:

. . iA : A → A#f H, iA(a) = a#1H , πA : A#f H → A, πA(a#h) = εH (h)a

. . iH : H → A#f H, iH (h) = 1A#h, πH : A#f H → H, πH (a#h) = εA(a)h for all a ∈ A, h ∈ H. It is straightforward to see that iH and πA are coalgebra maps while . iA and πH are Hopf algebra maps. The crossed product A#f H ﬁts into the following exact sequence:

iA . πH 0 / A / A#f H / H / 0

. such that πH : A#f H → H admits a section which is a coalgebra map and iA admits a retraction which is also a coalgebra map.

Recall that a Hopf subalgebra A of a Hopf algebra H is called normal if x(1)aS(x(2)) ∈ A and S(x1)ax(2) ∈ A for all x ∈ H, a ∈ A. We say that a Hopf algebra E factorizes through a Hopf subalgebra A and a subcoalgebra H if the multiplication map u : A ⊗ H → E, u(a ⊗ h) = ah, for all a ∈ A, h ∈ H is bijective. The next theorem describes the main property that characterizes the crossed product of Hopf algebras.

. Theorem 2.9.3 A Hopf algebra E is isomorphic as a Hopf algebra to a crossed product A#f H of Hopf algebras if and only if E factorizes through a normal Hopf subalgebra A and a subcoalgebra H such that 1E ∈ H.

∼ . . Proof: Suppose ﬁrst that E = A#f H. Then A#f H factorizes through A and H as the map . u : A ⊗ H → A#f H deﬁned by u(a ⊗ h) = iA(a)iH (h) = a#h is of course bijective. It . remains to prove that A is a normal Hopf subalgebra of A#f H. For any b ∈ A and a#h ∈ . A#f H we have:

114 2.9. CROSSED PRODUCT OF HOPF ALGEBRAS

S(a(1)#h(1))(b#1)(a(2)#h(2)) = h i = SA f(SH (h(2)), h(3)) #SH (h(1)) SA(a(1))#1 (b#1) = SA f SH (h(5), h(6)) SH (h(4)) . SA(a(1))b SH (h(3)) . a(2) f SH (h(2)), h(7) #SH (h(1))h(8) = SA f SH (h(5), h(6)) SH (h(4)) . SA(a(1))b SH (h(3)) . a(2) f SH (h(1))(1), h(7)(1) #SH (h(1))(2)h(7)(2) (2.106) = SA f SH (h(5), h(6)) SH (h(4)) . SA(a(1))b SH (h(3)) . a(2) f SH (h(1)), h(8) #SH (h(2))h(7) = SA f SH (h(4), h(5)) SH (h(3))(1) . SA(a(1))b SH (h(3))(2) . a(2) f SH (h(1)), h(7) #SH (h(2))h(6) (2.25) = SA f SH (h(4), h(5)) SH (h(3)) . SA(a(1))ba(2) f SH (h(1)), h(7)

#SH (h(2))h(6) (2.105) = SA f SH (h(3))(1), h(5)(1)) SH (h(2)) . SA(a(1))ba(2) f SH (h(1)), h(7)

#SH (h(3))(2)h(5)(2) (2.106) = SA f SH (h(3)), h(6)) SH (h(2)) . SA(a(1))ba(2) f SH (h(1)), h(7)

#SH (h(4))h(5) = SA f SH (h(3)), h(4)) SH (h(2)) . SA(a(1))ba(2) f SH (h(1)), h(5) #1 and the last line is obviously in A. We also have: (a(1)#h(1))(b#1)S(a(2)#h(2)) = a(1)(h(1) . b)#h(2) S(a(2)#h(3)) = a(1)(h(1) . b)#h(2) SA[f SH (g(4)), g(5) ]#SH (g(3)) SA(a(2))#1H = a(1)(h(1) . b) h(2) .SA[f SH (g(7)), g(8) ] f h(3),SH (h(6)) #h(4)SH (h(5)) SA(a(2))#1H = a(1)(h(1) . b) h(2) .SA[f SH (g(5)), g(6) ] f h(3),SH (h(4)) #1 SA(a(2))#1H

. the last line is again in A so it follows that A is a normal Hopf subalgebra in A#f H as desired. Conversely, assume now that a Hopf algebra E factorizes through a normal Hopf subalgebra A and a subcoalgebra H such that 1E ∈ H. It follows from Theorem 2.5.8 that E is isomorphic to

115 CHAPTER 2. UNIFIED PRODUCTS a uniﬁed product with the multiplication, the cocycle and the actions given by the formulas:

· : H ⊗ H → h, · := (εA ⊗ Id) ◦ ν

f : H ⊗ H → A, f := (Id ⊗ εH ) ◦ ν

. : H ⊗ A → A, . := (Id ⊗ εH ) ◦ µ

/ : H ⊗ A → H,/ := (εA ⊗ Id) ◦ µ where the coalgebra maps µ : H ⊗ A → A ⊗ H and ν : H ⊗ H → A ⊗ H are deﬁned by:

µ(h ⊗ a) := u−1(ha), ν(h ⊗ g) := u−1(hg) for all a ∈ A and h, g ∈ H. Since A is a normal Hopf subalgebra of E we get:

−1 µ : H ⊗ A → A ⊗ H, µ(h ⊗ a) := u (ha) = h(1)aS(h(2)) ⊗ h(3).

It follows from here that: h / a = εA(a)h, thus / is trivial and so the uniﬁed product is a crossed product of Hopf algebras.

Remark however that in the foregoing result the subcoalgebra H is in fact a Hopf algebra but not a Hopf subalgebra of E. If a Hopf algebra E factorizes through two Hopf subalgebras A and H, with A normal in E, then E is isomorphic as a Hopf algebra to a smash product as proved in [44, Proposition 2.5]. A classification result for crossed products of Hopf algebras can be derived from Theorem 2.6.4 where a more general approach to the classification problem, in the context of unified products, is given. This result is the counterpart at the level of Hopf algebras of the classical Schreier theorem on group extensions.

. .0 Proposition 2.9.4 Let A, H be two Hopf algebras and A#f H, A#f 0 H be two crossed products of Hopf algebras. The following are equivalent: . .0 (1) A#f H ≈ A#f 0 H (isomorphism of Hopf algebras, left A-modules and right H-comodules); (2) There exists a coalgebra lazy 1-cocyle u : H → A such that:

0 −1 (1) h . a = u (h(1))(h(2) . a)u(h(3))

0 −1 −1 (2) f (h, k) = u (h(1)) h(2) . u (k(1))f(h(3), k(2)) u(h(4)k(3)) for all a ∈ A and h, k ∈ H. In this case we shall say that the crossed systems (A, H, ., f) and (A, H, .0, f 0) are cohomologous and we denote the equivalence classes of crossed systems modulo such transformations by H2(H,A).

Remarks 2.9.5 1) A crossed system of Hopf algebras which is equivalent to the trivial one is called a coboundary. An example of a coboundary is given in Example 2.9.2, 4).

116 2.9. CROSSED PRODUCT OF HOPF ALGEBRAS

2) By applying Proposition 2.9.4 for the group Hopf algebras A = k[G] and H = k[G0], where G and G0 are two groups, we recover the general version of the classical Schreier theorem from group theory that classiﬁes all extensions of a group G by a group G0. For more details we refer to [19, pages 85-87]. In particular for G abelian we obtain the classical Schreier theorem (see [133, Theorem 7.34]).

The crossed product satisﬁes the following two universal properties.

. Proposition 2.9.6 Let A and H be two Hopf algebras and A#f H a crossed product of Hopf algebras. Then, we have:

(1) For any Hopf algebra X, any Hopf algebra map u : A → X and any coalgebra map v : H → X such that the following compatibilities hold for all h, g ∈ H, b ∈ A:

u f(h(1), g(1)) v(h(2)g(2)) = v(h)v(g) (2.108)

u(h(1) . b)v(h(2)) = v(h)u(b) (2.109)

. there exists a unique Hopf algebra map w : A#f H → X such that the following diagram commutes: . A#f H (2.110) i x; cGG i A xx GG H xx w GG xx GG xx G A u / XHo v

(2) For any Hopf algebra X, any Hopf algebra map v : X → H and any coalgebra map u : X → A such that the following compatibilities hold for all x, y ∈ X:

u(x(1)) ⊗ v(x(2)) = u(x(2)) ⊗ v(x(1)) (2.111) u(xy) = u(x(1)) v(x(2)) . u(y(1)) f v(x(3)), v(y(2)) (2.112)

. there exists a unique Hopf algebra map w : X → A#f H such that the following diagram commutes: X (2.113) G xx GG u xx GGv xx w GG xx GG {xx . G# A o A# H / H πA f πH

Proof: (1) Suppose ﬁrst that such a map w exists. Then, we have:

w(a#g) = w((a#1)(1#g)) = w(a#1)w(1#g) (2.110) = (w ◦ iA)(a)(w ◦ iH )(h) = u(a)v(g)

117 CHAPTER 2. UNIFIED PRODUCTS

. for all a#h ∈ A#f H and this proves the uniqueness of w. Next, we prove that w given by w(h#g) = u(a)v(g) is a Hopf algebra map such that diagram (2.110) commutes. w (a#h)(b#g) = w a(h(1) . b)f(h(2), g(1))#h(3)g(2) = u(a)u(h(1) . b)u f(h(2), g(1)) v(h(3)g(2)) (2.108) = u(a)u(h(1) . b)v(h(2))v(g) (2.109) = u(a)v(h)u(b)v(g) = w(a#h)w(b#g)

Thus w is an algebra map. The rest of the proof is straightforward and is left to the reader. . (2) Deﬁne w : X → A#f H by w(x) = u(x(1)) ⊗ v(x(2)). The uniqueness of w as well as the relations πH ◦ w = v and πA ◦ w = u are easy to establish. In what follows we prove that w is an algebra map. w(x)w(y) = u(x(1)) ⊗ v(x(2)) u(y(1)) ⊗ v(y(2)) = u(x(1)) v(x(2))(1) . u(y(1)) f v(x(2))(2), v(y(2))(1) #v(x(2))(3)v(y(2))(2) = u(x(1)) v(x(2)) . u(y(1)) f v(x(3)), v(y(2)) #v(x(4))v(y(3)) = u(x(1)(1)) v(x(1)(2)) . u(y(1)(1)) f v(x(1)(3)), v(y(1)(2)) #v(x(2))v(y(2)) (2.112) = u(x(1)y(1))#v(x(2))v(y(2))

= u(x(1)y(1))#v(x(2)y(2)) = w(xy)

Next, we are interested in which of the properties of the Hopf algebras A and H are preserved . by the crossed product A#f H. First, we should note that given the coalgebra structure of the . crossed product (i.e. the tensor product coalgebra) it is straightforward to see that A#f H is cocommutative (cosemisimple) if and only if both A and H are cocommutative (cosemisimple). Our next proposition extends the results obtained in [114] on the commutativity of crossed products of groups acting on rings. The cocycle f : H ⊗ H → A is called symmetric if f(g, h) = f(h, g) for all g, h ∈ H.

. . Proposition 2.9.7 Let A#f H be a crossed product of Hopf algebras. Then A#f H is commutative if and only if A and H are commutative, . is trivial and f is symmetric.

. Proof: Suppose ﬁrst that A#f H is commutative. Then, we have: a(h(1) . c)f h(2), g(1) #h(3)g(2) = c(g(1) . a)f g(2), h(1) #g(3)h(2) (2.114) for all a, c ∈ A and h, g ∈ H. By taking a = c = 1 and then applying ε ⊗ Id in (2.114) we get hg = gh for all h, g ∈ H that is, H is commutative. Moreover, if we apply Id⊗ε to the same equation (2.114) with a = c = 1 we get that f(h, g) = f(g, h) for all h, g ∈ H that is, f is symmetric.

118 2.9. CROSSED PRODUCT OF HOPF ALGEBRAS

Next, if we take c = 1 and h = 1 in (2.114) and then we apply Id ⊗ ε we obtain g . a = ε(g)a for all a ∈ A, g ∈ H, that is, . is trivial. Finally, A is commutative as a subalgebra in the . commutative Hopf algebra A#f H. Assume now that A and H are commutative Hopf algebras, . is trivial and f is symmetric. We then have: (a#h)(c#g) = acf h(1) g(1) #h(2)g(2) = caf g(1), h(1) #g(2)h(2) = (c#g)(a#h)

. . for all a#h, c#g ∈ A#f H. Thus A#f H is commutative.

. Proposition 2.9.8 Let A#f H be a crossed product of Hopf algebras.

. (1) If A#f H is involutory then both A and H are involutory Hopf algebras.

. (2) Suppose that H is cocommutative. Then A#f H is involutory if and only if A is involutory and g(1) . f SH (g(2)), g(3) = f g(1),SH (g(2)) for all g ∈ H.

. . Proof: (1) Since A#f H is involutory and A is a Hopf subalgebra of A#f H it follows that A is indeed involutory. Furthermore, we also have:

2 (2.107) 1#g = S (1#g) = S SA f(SH (g(2)), g(3)) #SH (g(1))

(2.107) h 2 2 ih 2 2 i = SA f SH (g(2)),SH (g(1)) #SH (g(3)) SA f(SH (g(4)), g(5)) #1 h 2 2 i 2 = SA f SH (g(2)),SH (g(1)) #SH (g(3)) f SH (g(4)), g(5) #1 2 2 2 = SA f SH (g(2)),SH (g(1)) SH (g(3)) . f SH (g(5)), g(6) #SH (g(4))

Hence, we obtained:

2 2 2 1#g = SA f SH (g(2)),SH (g(1)) SH (g(3)) . f SH (g(5)), g(6) #SH (g(4)) (2.115)

2 If we apply ε ⊗ Id in (2.115) we get SH (g) = g for all g ∈ H. . (2) Suppose ﬁrst that A#f H is involutory. It follows from (1) that A and H are involutory. Remark however that H is always involutory by the assumption of being cocommutative. Since . A#f H is involutory it follows again by (1) that relation (2.115) holds for all h ∈ H. By 2 applying Id ⊗ ε in (2.115) and having in mind that SH (g) = g for all g ∈ H, we get: SA f g(2),SH (g(1)) g(3) . f SH (g(4)), g(5) = ε(g) (2.116)

As H is cocommutative, (2.116) is equivalent to: SA f g(4),SH (g(1)) g(2) . f SH (g(3)), g(5) = ε(g) (2.117)

119 CHAPTER 2. UNIFIED PRODUCTS

Since f is a coalgebra map it follows, after inverting f in (2.117), that we have: g(1) . f SH (g(2)), g(3) = f g(1),SH (g(2)) for all g ∈ H. Assume now that A is involutory and g(1) . f SH (g(2)), g(3) = f g(1),SH (g(2)) (2.118) holds for all g ∈ H. As mentioned before, H is also involutory from the cocommutativity . assumption. Since T = {a#1H | a ∈ A}∪{1A#h | h ∈ H} generates A#f H as an algebra and S2(a#h) = S2(a#1)S2(1#h), we only need to prove that S2(a#1) = a#1 and S2(1#h) = 2 2 2 1#h for all a ∈ A, h ∈ H. Since S (a#1) = SA(a)#1 it follows trivially that S (a#1) = a#1 by the involutivity of A. As we already noticed in (1), we have:

2 S (1#g) = SA f g(2),SH (g(1)) g(3) . f SH (g(5)), g(6) #g(4) (2.105) = SA f g(2),SH (g(1)) g(4) . f SH (g(5)), g(6) #g(3) (2.118) = SA f g(2),SH (g(1)) f g(4),SH (g(5)) #g(3) = SA f g(3),SH (g(2)) f g(4),SH (g(1)) #g(5) = 1#g

. for all g ∈ H. Thus A#f H is involutory and the proof is ﬁnished.

. Remark 2.9.9 A more general result concerning the involutivity of the crossed product A#f H can be obtained by dropping the cocommutativity assumption on H. However the result is less transparent.

Our next result indicates a way of constructing integrals for the crossed products and proves that the crossed product of two Hopf algebras is semisimple if and only if both Hopf algebras are semisimple. This result is true regardless of the characteristic of the ﬁeld k (in characteristic 0 the result can be easily derived from the well-known result in [86]).

. Proposition 2.9.10 Let A#f H be a crossed product of ﬁnite dimensional Hopf algebras.

(1) If xA is a right integral in A and xH a right integral in H then xA#xH is a right integral . in A#f H;

1 2 . 1 2 1 2 (2) If Σt ⊗ t is a right integral in A#f H then zA := Σt ε(t ) and zH := Σε(t )t are right integrals in A respectively H;

. (3) A#f H is semisimple if and only if A and H are both semisimple.

120 2.9. CROSSED PRODUCT OF HOPF ALGEBRAS

. Proof: (1) Let a#h ∈ A#f H. We then have: (xA#xH )(a#h) = xA (xH )(1) . a f (xH )(2), h(1) #(xH )(3)h(2) = xAε (xH )(1) . a f (xH )(2), h(1) #(xH )(3)h(2) = ε(a)xAf (xH )(1), h(1) #(xH )(2)h(2) = ε(a)xAε f (xH )(1), h(1) #(xH )(2)h(2)

= ε(a)xA#xH h

= ε(a)ε(h)xA#xH

= ε(a#h)xA#xH where we used the fact that . and f are coalgebra maps and xA, xH are right integrals in A respectively H. 1 2 . (2) As Σt #t is a right integral in A#f H we have: 1 2 2 1 2 Σt (t(1) . a)f(t(2), h(1))#y(3)h(2) = Σε(a)ε(h)t #t (2.119) . for all a#h ∈ A#f H. By applying ε ⊗ Id in (2.119) and considering a#h := 1A#g we get, using the fact that f is a coalgebra map, that: Σε(t1)t2g = Σε(g)ε(t1)t2 for every g ∈ H. But the last equality is equivalent to: zH g = ε(g)zH . Hence zH is a right integral in H. Next, by 1 2 2 1 applying Id ⊗ ε in (2.119) and considering a#h := b#1H we get: Σbt ε(t ) = Σε(b)ε(t )t for all b ∈ A. But this is equivalent to bzA = ε(b)zA. Thus we obtained that zA is a right integral in A. (3) Assume ﬁrst that both A and H are semisimple algebras. It follows from Maschke’s theorem that there exist right integrals xA ∈ A and xH ∈ H such that ε(xA) = ε(xH ) = 1. From . (1) we have that xA#xH is a right integral in A#f H. Moreover, we have ε(xA#xH ) = . ε(xA)ε(xH ) = 1. Thus A#f H is semisimple. . 1 2 . Suppose now that A#f H is semisimple. Let Σt ⊗ t be a right integral in A#f H. Hence, 1 2 1 2 by (2), zA := Σt ε(t ) ∈ A and zH := Σε(t )t ∈ H are right integrals in A respectively H. 1 2 Since ε(Σt #t ) = 1 it follows that we also have ε(zA) = ε(zH ) = 1. Hence A and H are both semisimple algebras.

To end with, we describe, as an easy consequence of Theorem 2.8.6, the coquasitriangular or braided structures on the crossed product of Hopf algebras. In other words, we determine all . braided structures that can be deﬁned on the monoidal category of A#f H - comodules. The following notions are special cases of those introduced at the beginning of Section 2.8 by considering / to be the trivial action.

Deﬁnition 2.9.11 Let A, H be two Hopf algebras, f : H ⊗ H → A a coalgebra map and p : A ⊗ A → k a braiding on A. A linear map u : A ⊗ H → k is called (p,f) - right skew pairing on (A, H) if the following compatibilities are fulﬁlled for any a, b ∈ A, g, t ∈ H:

(RS1) u(ab, t) = u(a, t(1))u(b, t(2))

121 CHAPTER 2. UNIFIED PRODUCTS

(RS2) u(1, h) = ε(h)

(RS3) u(a(1), g(2)t(2))p a(2), f(g(1), t(1)) = u(a(1), t)u(a(2), g)

(RS4) u(a, 1) = ε(a)

Deﬁnition 2.9.12 Let A, H be two Hopf algebras, f : H ⊗ H → A a coalgebra map and p : A ⊗ A → k a braiding on A. A linear map v : H ⊗ A → k is called (p,f) - left skew pairing on (H,A) if the following compatibilities are fulﬁlled for any b, c ∈ A, h, g ∈ H: (LS1) p f(h(1), g(1)), c(1) v(h(2)g(2), c(2)) = v(h, c(1))v(g, c(2))

(LS2) v(1, a) = ε(a)

(LS3) v(h, bc) = v(h(1), c)v(h(2), b)

(LS4) v(h, 1) = ε(h)

Examples 2.9.13 1) If f = εH ⊗ εH then the notions of (p,f) - right skew pairing and (p,f) - left skew pairing coincide with the notion of skew pairing on (A, H) respectively (H,A). 2) Let L =< t | tn = 1 > and G =< g | gm = 1 > be two cyclic groups of orders n respectively m and consider the group Hopf algebras A = k[L] and H = k[G]. In this setting a coalgebra map f : k[G] ⊗ k[G] → k[L] is completely determined by a map α : {0, 1, ..., m − 1} × {0, 1, ..., m − 1} → {0, 1, ..., n − 1} such that f(gi, gj) = tα(i, j). Recall that the braidings on a group Hopf algebra k[L] are in one-to-one correspondence with the bicharacters on L. More precisely, the braidings on k[L] are given by:

p : k[L] ⊗ k[L] → k, p(ta, tb) = τ ab

where a, b ∈ 0, n − 1 and τ ∈ k such that τ n = 1. Then there exists u : k[L] ⊗ k[G] → k a (p, f) - right skew pairing if and only if α is a symmetric map and there exists υ ∈ k such that υn = 1 and υm = τ α(1, m−1). In this case the (p, f) - right skew pairing u : k[L] ⊗ k[G] → k is given by: u(ta, gb) = υabτ −α(1, b−1), a ∈ 0, n − 1, b ∈ 0, m − 1.

Deﬁnition 2.9.14 Let A and H be two Hopf algebras, f : H ⊗ H → A a coalgebra map, p : A ⊗ A → k a braiding on A, u : A ⊗ H → k a (p, f) - right skew pairing on (A, H) and v : H ⊗ A → k a (p, f) - left skew pairing on (H,A). A linear map τ : H ⊗ H → k is called (u, v) - skew braiding on H if the following compatibilities are fulﬁlled for all h, g, t ∈ H: (SBR1) u f(h(1), g(1)), t(1) τ(h(2)g(2), t(2)) = τ(h, t(1))τ(g, t(2))

122 2.9. CROSSED PRODUCT OF HOPF ALGEBRAS

(SBR2) τ(1, h) = ε(h)

(SBR3) τ(h(1), g(2)t(2))v h(2), f(g(1), t(1)) = τ(h(1), t)τ(h(2), g)

(SBR4) τ(g, 1) = ε(g)

(SBR5) τ(h(1), g(1))h(2)g(2) = g(1)h(1)τ(h(2), g(2))

Remark that if f = εH ⊗ εH then the (u, v) - skew braiding τ is a regular braiding on H.

Theorem 2.9.15 Let (A, H, ., f) be a crossed system of Hopf algebras. The following are equivalent:

. 1) (A#f H, σ) is a braided Hopf algebra

2) There exist four linear maps p : A ⊗ A → k, τ : H ⊗ H → k, u : A ⊗ H → k, v : H ⊗ A → k such that (A, p) is a braided Hopf algebra, u is a (p, f) - right skew pairing on (A, H), v is a (p, f) - left skew pairing on (H,A), τ is a (u, v) - skew braiding on H and the following compatibilities are fulﬁlled:

v(h(1), b(1))(h(2) . b(2)) ⊗ h(3) = b(1) ⊗ h(1)v(h(2), b(2)) (2.120)

(g(1) . a(1)) ⊗ g(2)u(a(2), g(3)) = u(a(1), g(1))a(2) ⊗ g(2) (2.121)

τ(h(1), g(1))f(h(2), g(2)) = f(g(1), h(1))τ(h(2), g(2)) (2.122)

u(a(1), g(2))p(a(2), g(1) . c) = p(a(1), c)u(a(2), g) (2.123)

τ(h(1), g(2))v(h(2), g(1) . c) = v(h(1), c)τ(h(2), g) (2.124)

p(h(1) . b, c(1))v(h(2), c(2)) = v(h, c(1))p(b, c(2)) (2.125)

u(h(1) . b, t(1))τ(h(2), t(2)) = τ(h, t(1))u(b, t(2)) (2.126)

. . and the braiding σ :(A#f H) ⊗ (A#f H) → k is given by:

σ(a#h, b#g) = u(a(1), g(1))p(a(2), b(1))τ(h(1), g(2))v(h(2), b(2)) (2.127)

for all a, b, c ∈ A and h, g, t ∈ H.

Proof: It follows from Theorem 2.8.6 by considering / to be the trivial action.

The next Corollary gives necessary and sufﬁcient conditions for a smash product Hopf algebra to be braided. The result can also be derived from [78, Theorem 3.4] by considering T : H ⊗ B → B ⊗ H given by T (h ⊗ b) = h(1) . b ⊗ h(2), where B is a left H-module bialgebra via ..

123 CHAPTER 2. UNIFIED PRODUCTS

Corollary 2.9.16 Let (A, H, ., f) a crossed system of Hopf algebras such that f is trivial. The following are equivalent:

1) (A#. H, σ) is a braided Hopf algebra

2) There exist four linear maps p : A ⊗ A → k, τ : H ⊗ H → k, u : A ⊗ H → k, v : H ⊗ A → k such that (A, p) and (H, τ) are braided Hopf algebras, u and v are skew pairings, the compatibilities (2.120)-(2.121) and (2.123)-(2.126) are fulﬁlled and the braiding σ is given by (2.127).

Proof: It is straightforward to see that if f is the trivial cocycle then relation (2.122) from Theorem 2.9.15 is trivially fulﬁlled while relations (RS3) and (LS1) which are satisﬁed by u and v collapse to (BR3) respectively (BR1). Corollary 2.9.17 Let (A, H, ., f) be a crossed system of Hopf algebras with A a commutative Hopf algebra, . the trivial action and (H, τ) a braided Hopf algebra such that:

τ(h(1), g(1))f(h(2), g(2)) = f(g(1), h(1))τ(h(2), g(2)) for all h, g ∈ H. Then A#. H is a braided Hopf algebra with the braiding given by: σ(a#h, b#g) = ε(a)ε(b)τ(h, g) for all h, g ∈ H.

Proof: Since A is commutative we can consider on A the trivial braiding given by p = εA ⊗ εA. Moreover u = εA ⊗ εH is a (p, f) - right skew pairing on (A, H) and v = εH ⊗ εA is a (p, f) - left skew pairing on (H,A). In this case, an (u, v) - skew braiding on H is actually a regular braiding and the conclusion follows by Theorem 2.9.15. Corollary 2.9.18 Let A and H be Hopf algebras. The following are equivalent:

1) (A ⊗ H, σ) is a braided Hopf algebra

2) There exist four linear maps p : A ⊗ A → k, τ : H ⊗ H → k, u : A ⊗ H → k, v : H ⊗ A → k such that (A, p) and (H, τ) are braided Hopf algebras, u and v are skew pairings on (A, H) respectively (H,A) and the following compatibilities are fulﬁlled:

v(h(1), b(1))b(2) ⊗ h(2) = b(1)v(h(2), b(2)) ⊗ h(1)

a(1) ⊗ g(1)u(a(2), g(2)) = a(2) ⊗ u(a(1), g(1))g(2)

u(a(1), g)p(a(2), c) = p(a(1), c)u(a(2), g)

τ(h(1), g)v(h(2), c) = v(h(1), c)τ(h(2), g)

p(b, c(1))v(h, c(2)) = v(h, c(1))p(b, c(2))

u(b, t(1))τ(h, t(2)) = τ(h, t(1))u(b, t(2)) and the braiding σ is given by (2.127).

124 2.9. CROSSED PRODUCT OF HOPF ALGEBRAS

Proof: It follows from Corollary 2.9.16 by considering h . a = ε(h)a.

Remark 2.9.19 It is easy to see that if, for example, A and H are both cocommutative then the compatibilities in Corollary 2.9.18 are trivially fulﬁlled.

To end with we construct some explicit braidings on the crossed product of Hopf algebras.

Examples 2.9.20 1) As a ﬁrst example we describe, using Corollary 2.9.18, the braidings on the tensor product k[X] ⊗ k[X], where k[X] is the polynomial algebra. Recall that k[X] is a commutative and cocommutative Hopf algebra with the coalgebra structure and antipode given by:

∆(X) = X ⊗ 1 + 1 ⊗ X, ε(X) = 0,S(X) = −X

Any element α ∈ k, induces a skew pairing ϕ on k[X] as follows:

0, if i 6= j ϕ(Xi,Xj) = i!αi, if i = j

Moreover, since k[X] is commutative and cocommutative the braidings on k[X] coincide with the skew pairings on (k[X], k[X]). Let p, τ, u and v be the braidings induced by α, β, γ, τ ∈ k. Then, according to Corollary 2.9.18 a braiding σ on the tensor product k[X] ⊗ k[X] is given by:

min{a,d} X a b c d σ(Xa ⊗ Xb,Xc ⊗ Xd) = (c − a + i)! i d − i a − i i i=max{0,a−c} (a − i)!(d − i)!i!αa−iβd−iγiτ c−a+i if a + b = c + d and σ(Xa ⊗ Xb,Xc ⊗ Xd) = 0 for a + b 6= c + d. Moreover, all braidings on k[X] ⊗ k[X] arise in this way. 3 2) In what follows k is a ﬁeld such that 2 is invertible in k. Let H = k[C3] = k < a | a = 1 > be the group Hopf algebra and A = H4 be Sweedler’s Hopf algebra. Recall that H4 is generated as an algebra by two elements g and x subject to relations:

g2 = 1, x2 = 0, xg = −gx while the coalgebra structure and antipode are given by:

∆(g) = g ⊗ g, ∆(x) = g ⊗ x + x ⊗ 1, ε(g) = 1

ε(x) = 0,S(g) = g, S(x) = −gx. It is a straightforward computation to see that A and H together with the maps . : H ⊗ A → A and f : H ⊗ H → A deﬁned below is a crossed system of Hopf algebras:

f(a, a) = f(a2, a2) = g and f(ai, aj) = 1 for (i, j) ∈/ {(1, 1), (2, 2)}

125 CHAPTER 2. UNIFIED PRODUCTS

a . g = a2 . g = g, a . x = a2 . x = −x, a . gx = a2 . gx = −gx

. In order to describe the braidings on the crossed product H4#f k[C3] we start by listing a braided structure on H4, left/right skew pairings and skew braidings. For any α ∈ k, (H4, p) is a braided Hopf algebra, where p : H4 ⊗ H4 → k is given by:

p 1 g x gx 1 1 1 0 0 g 1 -1 0 0 x 0 0 α α gx 0 0 α α

The linear map u : H4⊗k[C3] → k deﬁned below is a (p, f) - right skew pairing on (H4, k[C3]):

u(g, a) = u(g, a2) = −1

u(x, a) = u(x, a2) = u(gx, a) = u(gx, a2) = 0

The linear map v : k[C3] ⊗ H4 → k deﬁned below is a (p, f) - left skew pairing on (k[C3],H4):

u(a, g) = u(a2, g) = −1

u(a, x) = u(a2, x) = u(a, gx) = u(a2, gx) = 0

3 For any γ ∈ k such that γ = 1, the linear map τ : k[C3] ⊗ k[C3] → k deﬁned below is a (u, v) - skew braiding on k[C3]:

τ 1 a a2 1 1 1 1 a 1 γ −γ2 a2 1 −γ2 −γ

Moreover, the maps p, u, v and τ satisfy conditions (2.120)-(2.126) in Theorem 2.9.15. Thus, σ : . . . H4#f k[C3] ⊗ H4#f k[C3] → k is a braiding on the crossed product H4#f k[C3], where:

σ(b ⊗ y, c ⊗ z) = u(b(1), z(1))p(b(2), c(1))τ(y(1), z(2))v(y(2), c(2)) is given by:

126 2.9. CROSSED PRODUCT OF HOPF ALGEBRAS

σ 1#1 1#a 1#a2 g#1 g#a g#a2 1#1 1 1 1 1 1 1 1#a 1 γ −γ2 −1 −γ γ2 1#a2 1 −γ2 −γ −1 γ2 γ g#1 1 −1 −1 −1 1 1 g#a 1 −γ γ2 1 −γ γ2 g#a2 1 γ2 γ 1 γ2 γ x#1 0 0 0 0 0 0 x#a 0 0 0 0 0 0 x#a2 0 0 0 0 0 0 gx#1 0 0 0 0 0 0 gx#a 0 0 0 0 0 0 gx#a2 0 0 0 0 0 0

σ x#1 x#a x#a2 gx#1 gx#a gx#a2 1#1 0 0 0 0 0 0 1#a 0 0 0 0 0 0 1#a2 0 0 0 0 0 0 g#1 0 0 0 0 0 0 g#a 0 0 0 0 0 0 g#a2 0 0 0 0 0 0 x#1 α −α −α −α α α x#a α −αγ αγ2 α −αγ αγ2 x#a2 α αγ2 αγ α αγ2 αγ gx#1 α α α α α α gx#a α αγ −αγ2 −α −αγ αγ2 gx#a2 α −αγ2 −αγ −α αγ2 αγ

Bibliographical Notes

Section 2.2, Section 2.4, Section 2.5 and Section 2.6 are inspired on the joint work with G. Mili- taru [12]. Section 2.3 is deduced from the author’s joint work with G. Militaru [16]. Section 2.7 is taken from the author’s joint work with G. Militaru [15]. Section 2.8 is taken from the author’s paper [6]. Section 2.9 is contained in the author’s paper [7].

127 CHAPTER 2. UNIFIED PRODUCTS

128 3 Classifying bicrossed products of quantum groups. Deformations of a Hopf algebra and descent type theory

3.1 Motivating problems

The aim of this chapter is to prove that there exists a very rich theory behind the so-called bicrossed product (or double cross product in Majid’s terminology) of two objects arising from the factorization problem which deserves to be developed further mainly because of the major impact they have in at least three different problems: the classification of objects of a given dimension, the development of a general descent type theory for a given extension A ⊆ E (including a deformation type theory as a subsequent problem) which we called bicrossed descent theory as the converse of the factorization problem and also the development of some new types of cohomologies that will be the key players for both problems. All results presented below provide a detailed answer at the level of Hopf algebras for the three problems mentioned above and offer an argument for the major role that bicrossed products can play. In particular, for finite quantum groups we describe a new way of approaching the classification problem which we hope to be effective in the future. In order to maintain a general frame for our discussion, considering that bicrossed products were introduced and studied in various areas of mathematics, we will consider C a category whose objects are sets endowed with various algebraic, topological, differential structures. To illustrate, we can thing of C as the category of groups, groupoids or quantum groupoids, algebras, Hopf algebras, local compact groups or local compact quantum groups, Lie groups, Lie algebras and so on. Let A and H be two given objects of C. We say that an object E ∈ C factorizes through A and H if E can be written as a ’product’ of A and H, where A and H are viewed as subobjects of E having minimal intersection. Here, the ’product’ depends on the nature of the category. For instance, if C = Gr, the category of groups, then a group E factorizes through two subgroups A and H if E = AH and A ∩ H = {1}. This is called in group theory an exact factorization of E and can be restated equivalently as the fact that the multiplication map A×H → E, (a, h) 7→ ah is bijective. The last assertion is also taken as a definition of factorization for other categories like: algebras [49], Hopf algebras [100], Lie groups or Lie algebras [96], [84], [103], locally

129 CHAPTER 3. CLASSIFYING BICROSSED PRODUCTS OF QUANTUM GROUPS compact quantum groups [145] and so on. The factorization problem is then the following very natural and elementary question: The factorization problem. Let A and H be two given objects of C. Describe and classify up to an isomorphism all objects of C that factorize through A and H. There is also an interesting converse of the above problem, called here the bicrossed descent theory, which we introduce below having as main source of inspiration the classical descend theory for modules [83]. Now let A ⊂ E be two given objects of C such that A is an subobject of E.A factorization A-form of E is a suboject H of E such that E factorizes through A and H. We denote by F(A, E) the (possibly empty) full subcategory of C of all factorization A-forms of E. The bicrossed descent theory consists of the following two questions: Existence of forms. Let A ⊂ E be an extension in C. Does there exist a factorization A-form of E? Description and classification of forms. If a factorization A-form of E exists, describe and classify up to isomorphism all factorization A-forms of E. We will now provide a historical background of the three problems. First we consider the trivial case in which the category C = Ab, where Ab is the category of abelian groups or more general the category of left R-modules over a ring R. In this case the three problems become trivial in the following way: an abelian group E factorizes through two subgroups A and H if and only if E =∼ A ⊕ H, that is E is the coproduct of A and H. Indeed, the direct summands in an abelian group E are either kernels or images of idempotent endomorphisms of E, that is A =∼ Ker(f), H =∼ Im(f), for some f = f 2 ∈ End(E). An abelian group (or more general a left R-module) which has no proper decompositions is called indecomposable. We also recall the existence of the well known Krull-Schmidt-Azumaya theorem as a tool related to the uniqueness of these decompositions. Now we take a step forward and consider C = Gr, the category of groups. Here things change radically: all three problems become difficult. The factorization problem for groups was formulated by O. Ore [115] but its roots are much older and descend to E. Maillet’s 1900 paper [94]. It can be seen as the dual of the more famous extension problem of O. L. Holder.¨ Moreover, little progress has been made on this problem. For instance, in the case of two cyclic groups A and H, not both finite, the problem was started by L. Redei´ in [132] and finished by P.M. Cohn in [56], without the classification part. If A and H are both finite cyclic groups the problem is more difficult and seems to be still an open question, even though J. Douglas [65] has devoted four papers to the subject. In [57] all groups that factorize trough an alternating group or a symmetric group are completely described. One of the famous results about the factorization problem for groups remains Ito’s theorem [75] proved in the 50’s: any product of two abelian groups is a metabelian group. The problem concerning the existence of forms for groups was intensively studied in group theory in his equivalent form: given a group E find all exact factorizations of it. Starting with the 1980’s various papers dealing with this problem were written (see [31], [68], [88], [89], [126], [146] and the references therein). The problem of describing and classifying forms at the level of groups was recently posed by Kuperberg [85]: let E be a group that has two exact

130 3.1. MOTIVATING PROBLEMS factorizations E = AH = AH0. What is the relation between the groups H and H0? This was our starting point in formulating the third problem above, as well as the classical descend theory [83] and the one for Hopf algebras [121]. Going back to the factorization problem for groups, an important step in dealing with this problem was the construction of the bicrossed product A ./ H associated to a matched pair (A, H, ., /) of groups given by Takeuchi [143]. A group E factorizes through two subgroups A and H if and only if there exists a matched pair of groups (A, H, ., /) such that E =∼ A ./ H. Thus the factorization problem can be restated in a pure computational manner: Let A and H be two given groups. Describe the set of all matched pairs (A, H, ., /) and classify up to an isomorphism all bicrossed products A ./ H. This is a strategy that has to be followed for the factorization problem in any category C. In fact, the bicrossed product construction at the level of groups served as a model for similar constructions in other fields of mathematics. The first step was made by Majid in [100, Proposition 3.12] where a twisted version of bicrossed product of two Hopf algebras associated to a matched pair was introduced, under the name of double cross product, whose purpose was not the factorization problem but to give an elegant description for the Drinfel’d double of a finite dimensional Hopf algebra. Then the construction was performed for Lie Algebras [96], [103] and Lie groups [96], [84], algebras [49] or more general in [55], coalgebras [48], groupoids [18], [95], locally compact groups [30] or locally compact quantum groups [145], and so on. At the level of algebras the bicrossed product of two algebras is usually called in the literature the twisted tensor product algebra. In conclusion, if we are only looking for the description part of the factorization problem, formulated in an arbitrary category C, the following general principle (for the categories mentioned above this principle becomes a theorem) should work: an object E ∈ C factorizes through A and H if and only if E =∼ A ./ H, where A ./ H is a ’bicrossed product’ in the category C associated to a ’matched pair’ between the objects A and H. The classification part of the factorization problem is now clear: it consists of classifying the bicrossed products A ./ H associated to all matched pairs between A and H. This is the strategy that we follow for the category of Hopf algebras. For other categories, the steps taken into this direction are still shy, including the group case as well as the algebras case presented above. The problem of classifying bicrossed products of two algebras started with [48, Examples 2.11] where all bicrossed product between two group algebras of dimension two are completely described and classified. For recent results related to the classification of bicrossed product of two algebras we refer to [90], [91], [70]. On the other hand, a step forward in the third problem related to the description of forms, which we called the deformation of forms was proved at the level of algebras under the name of invariance under twisting theorem [76, Theorem 4.4]. The problem of classifying up to isomorphism all Hopf algebras of a given dimension is one of the central themes in Hopf algebra theory. There are complete classifications for Hopf algebras of a given dimension such as p, 2p, p2, 2p2 (see [110], [72], [147]), p a prime. However, there are no standard methods of tackling this problem, in all the cases mentioned above ad-hoc arguments are used. The only method for classifying a class of Hopf algebras, namely those whose coradical is a Hopf algebra, is the lifting method, developed in [22]. Following Holder’s

131 CHAPTER 3. CLASSIFYING BICROSSED PRODUCTS OF QUANTUM GROUPS model from group extensions, this method relies on the theory of Hopf algebra extensions and the celebrated Radford biproduct. In this chapter we introduce a new way of classifying Hopf algebras using bicrossed products instead of extension theory for Hopf algebras and Radford’s biproduct. The present chapter offers an answer to the first and the third problem above if C = Hopf, the category of Hopf algebras. The second problem, namely the existence of forms, needs to be treated ”case by case” for every given Hopf algebra extension A ⊆ E, a computational part of it can not be avoided. This was also the approach used in the similar problem at the level of groups, i.e. corresponding to the Hopf algebra extension k[A] ⊆ k[G], for two groups A and G with A ≤ G. The problem of finding all factorizations of a given group G was started in the 80’s and there is a very rich literature on the subject: see for instance [11], [12], [31], [68], [88], [89], [126] and the references therein. The chapter is organized as follows. In Section 3.1 we shall recall the basic concepts that will be used throughout the chapter. First we briefly recall the theory of forms and classical descent theory for Hopf algebras following [121] as is very much related to our bicrossed descent theory. The twisted version of bicrossed product associated to a matched pair of Hopf algebras (A, H, ., /) was introduced by Majid [100, Proposition 3.12] under the name of double crossproduct. We will review this construction in Theorem 3.1.1 by considering the bicrossed product as a special case of the more general unified product introduced in the previous chapter for two reasons: on the one hand this approach provides us with the advantage of obtaining the converse of [81, Theorem IX 2.3] or [101, Theorem 7.2.2] and on the other hand it will be used in the bicrossed descent theory in Section 3.4. Majid’s result in Theorem 3.1.3 proves that the factorization problem for Hopf algebras can be restated in a computational manner exactly as in the group case: Let A and H be two given Hopf algebras. Describe the set of all matched pairs (A, H, ., /) and classify up to an isomorphism all bicrossed products A ./ H. Section 3.2 is devoted to proving some purely technical results which will be intensively used throughout the chapter. Theorem 3.2.2 describes completely the set of all morphisms of Hopf algebras ψ : A ./ H → A0 ./0 H0 between two arbitrary bicrossed products. In particular, the set of all Hopf algebra maps between two semi-direct (or smash) products of Hopf algebras is described in Corollary 3.2.3. This result gives the first application: the parametrization of all Hopf algebra morphisms between two Drinfel’d doubles associated to two finite groups G and H is given in Corollary 3.2.4. In particular, if G = H, the group EndHopf D(k[G]) of all Hopf algebras endomorphism is fully described. Section 3.3 deals with the classification part of the factorization problem for which the group 1 Hl,c(H,A) of all coalgebra lazy 1-cocycles of H with coefficients in A introduced in Defini- tion 2.6.3 plays the crucial role. Let A and H be two given Hopf algebras. Theorem 3.3.7 is the classification theorem for bicrossed products: all Hopf algebras E that factorize through A and H are classified up to an isomorphism that stabilizes A by a cohomological type object 2 1 H (A, H) in the construction of which the key role is played by pairs (r, v) ∈ Hl,c(H,A) × 1 Aut CoAlg(H), consisting of a coalgebra lazy 1-cocycle r : H → A and an unitary automorphism of coalgebras v : H → H related by a certain compatibility condition. The classification of bicrossed products up to an isomorphism that stabilizes one of the terms has at least two

132 3.1. MOTIVATING PROBLEMS strong motivations: the first one is the cohomologically point of view which descends to the classification theory of Holders’s groups extensions [133, Theorem 7.34] and the second one is the problem of describing and classifying the A-forms of a Hopf algebra from descent theory. We indicate only two of the several applications of Theorem 3.3.7: Corollary 3.3.8 gives neces- 0 sary and sufficient conditions for two generalized quantum doubles Dλ(A, H) and Dλ0 (A, H ) to be isomorphic as Hopf algebras and left A-modules. Corollary 3.3.9 is a Schur-Zassenhaus type theorems for the bicrossed product of Hopf algebras: it provides necessary and sufficient conditions for a bicrossed product A ./ H to be isomorphic to a semi-direct product A#0H0 such that the isomorphism stabilizes A. Section 3.4 offers the full answer to the third problem above on the description and classification of forms as part of what we have called the bicrossed descent theory. The answer will be given in four steps, each of them of interest in its own right, as follows: deformation of a Hopf algebra, deformations of forms, the description of forms and finally the classification of forms. In Theorem 3.4.7 a general deformation of a given Hopf algebra H is introduced. This deformation Hr of H is associated to an arbitrary matched pair of Hopf algebras (A, H, ., /) and to an (., /)-cocycle r : H → A in the sense of Definition 3.4.4. As a coalgebra Hr = H, with the new multiplication • defined by h • g := h / r(g(1)) g(2) for all h, g ∈ Hr = H. Then Hr is a new Hopf algebra called the r-deformation of H. At this point we should notice that there exists two other famous deformations of a given Hopf algebras in the literature. The first one was introduced by Drinfel’d [66]: the comultiplication of a Hopf algebra H is deformed using an invertible element R ∈ H ⊗ H, called twist, in order to obtain a new Hopf algebra HR. As an algebra HR is equal to H, but the comultiplication is given by the formula ∆R(h) := R∆(h)R−1, for all h ∈ H. In order for HR to be a Hopf algebra, R has to satisfy some compatibility conditions which are equivalent to saying that R−1 is a cocycle in the Harrison cohomology. Drinfel’d twist deformation proved itself to be a key tool in the classification theory of finite dimensional semisimple Hopf algebras as well as in the representation theory of fusion categories. The dual case was introduced by Doi [60]: the algebra structure of a Hopf algebra H was deformed using a Sweedler cocycle σ : H ⊗ H → k as follows: let Hσ = H, as a coalgebra, with the new multiplication given by

−1 h · g := σ(h(1), g(1)) h(2)g(2) σ (h(3), g(3)) for all h, g ∈ H. Then Hσ is a new Hopf algebra [60, Theorem 1.6] and among several interesting applications in quantum groups we mention that the Drinfel’d double D(H) is a special case of this deformation [63, Remark 2.3]. Now let A ⊆ E be an extension of Hopf algebras and F(A, E) the small category, possibly empty, of all factorization A-forms of E: hence, F(A, E) is the category of all Hopf subalgebras H ⊆ E such that E factorizes through A and H. Let F sk(A, E) be the skeleton of F(A, E), that is a set of types of isomorphisms of all factorization A-forms of E. The factorization index of the extension E/A, introduced in Deﬁnition 3.4.2 and denoted by [E : A]f , is the cardinal of F sk(A, E), i.e. [E : A]f = | F sk(A, E) |. The extension A ⊆ E is called rigid if [E : A]f = 1.

133 CHAPTER 3. CLASSIFYING BICROSSED PRODUCTS OF QUANTUM GROUPS

Rigid extensions are interesting for the following reason. Assume that E/A is rigid: if E =∼ A ./ H =∼ A ./0 H0, then H =∼ H0. This is a Krull-Schmidt-Azumaya type theorem for bicrossed product of two Hopf algebras: the rigid extensions of Hopf algebras are exactly those for which the decomposition as a bicrossed product is unique. Examples of rigid extensions as well as of extensions E/A such that [E : A]f ≥ 2, which are quite rare, are provided. Theorem 3.4.9 proves that if r : H → A is an (., /)-cocycle, then the r-deformation Hr is a factorization A- r form of the bicrossed product A ./ H, that is there exists a new matched pair (A, Hr,. ,/) such r ∼ that A ./ Hr = A ./ H. We called this result deformation of forms: another name used for a similar result at the level of algebras is invariance under twisting theorem [76, Theorem 4.4]. The description of forms is given in Theorem 3.4.10 which is the converse of Theorem 3.4.9: if H is a given factorization A-form of E then any other form H is isomorphic as a Hopf algebra with an Hr, for some (., /)-cocycle r : H → A. This result is interesting in its own right as it proves that in order to find all the objects of the category F(A, E) of all factorization A-forms of E it is enough to know only one object H: all other objects are deformations of H. Finally, as a conclusion of these theorems, the classification of forms is proved as the main result of the section, namely Theorem 3.4.6: if H is a given factorization A-form of E then there exists a bijection between the set of types of isomorphisms of all factorization A-forms of E and a new cohomological object HA2(H,A | (., /)). In particular, we obtain a formula for computing the factorization index of a given extension A ⊆ E: [E : A]f = |HA2(H,A | (., /))|. Mutatis- mutandis, Theorem 3.4.6 can be viewed as a bicrossed version for Hopf algebras of the classical result in descent theory: if k ⊆ l is a faithfully flat extension of commutative rings then the Amitsur cohomology group is isomorphic to the relative Picard group Pic(l/k) ([83]). All the results proved in Section 3.4 can be translated to groups by replacing the category C = Hopf with the category Gr of groups, without needing a proof. We write down briefly these new results at the end of the section. In Section 3.5 we provide some explicit examples in full details: for two given Hopf algebras A and H we will describe by generators and relations and classify up to an isomorphism all Hopf algebras E that factorize through A and H. Furthermore, for any such Hopf algebra E the factorization index [E : A]f is computed. There are three steps that we go through: first of all we compute the set of all matched pairs between A and H. This is the computational part of our schedule and can not be avoided. Then we describe by generators and relations the bicrossed products A ./ H associated to all these matched pairs. Finally, using Theorem 3.2.2, we shall classify up to an isomorphism these bicrossed products A ./ H. As an application, the group Aut Hopf (A ./ H) of all Hopf algebra automorphisms of a given bicrossed product is computed.

Let k be a ﬁeld of characteristic 6= 2 and H4 the Sweedler four dimensional Hopf algebra. For a positive integer n, let Cn be the cyclic group of order n generated by c and Un(k) = {ω ∈ n k | ω = 1} the cyclic group of n-th roots of unity in k of order ν(n) = |Un(k)|. The group Un(k) depends heavily on the base ﬁeld k. Proposition 3.5.3 proves that Un(k) parameterizes the set of all matched pairs (H4, k[Cn], ., /): i.e. there exists a bijective correspondence between the set of all matched pairs (H4, k[Cn], ., /) and the group Un(k). Corollary 3.5.4 shows that a Hopf ∼ algebra E factorizes through H4 and k[Cn] if and only if E = H4n, ω, where for any ω ∈ Un(k), the quantum group at root of unity H4n, ω is described explicitly by generators and relations. It is a non-commutative non-cocommutative, pointed and non-semisimple 4n-dimensional Hopf

134 3.1. MOTIVATING PROBLEMS

algebra. Furthermore, the canonical extension H4 ⊆ H4n, ω is rigid. Theorem 3.5.7, and more precisely Theorem 3.5.10, describe precisely the number of types of isomorphisms of this family of Hopf algebras {H4n, ω | ω ∈ Un(k)}. The beauty of this classification result is given by the Dirichlet’s theorem on primes in an arithmetical progression which was used in a key step in α1 αr proving Theorem 3.5.10. Let ν(n) = p1 ··· pr be the prime decomposition of ν(n). If ν(n) is odd, then the number of types of such Hopf algebras is (α1 + 1)(α2 + 1) ··· (αr + 1). On the α1 α2 αr other hand, if ν(n) = 2 p2 ··· pr is even, then the number of types of such Hopf algebras is α1(α2 + 1) ··· (αr + 1). As an application, the infinite abelian group Aut Hopf (H4n, ω) of Hopf algebra automorphisms of H4n, ω is described in Corollary 3.5.14. Theorem 3.5.16 describes and classifies all Hopf algebras that factorize through two Sweedler’s Hopf algebras. These are the following: H4 ⊗H4, D(H4) the Drinfel’d double of H4, and a new Hopf algebra which we be denoted by H16, 0, described in detail by generators and relations. This example reveals yet another very interesting side of this theory. Let H be a finite dimensional Hopf algebra. Then the Drinfel’d double D(H) factorizes through (H∗)cop and H and thus D(H) = (H∗)cop ./ H, as proved by Majid (Example 3.1.2). Theorem 3.5.16 shows that, besides the Drinfel’d double D(H), there may be other quantum groups which factorize through (H∗)cop and H. Proposition 3.5.15 presents another type of border case: it shows that we can have various matched pairs for two given Hopf algebras, but all bicrossed products associated to these matched pairs are isomorphic to the trivial one, namely the tensor product of Hopf algebras. Such an example is obtained for H4 and k[C2 × C2], the group algebra of the Klein’s group.

Notational conventions

Throughout this chapter, k will be a field. For a positive integer n we denote by Un(k) = {ω ∈ n k | ω = 1} the cyclic group of n-th roots of unity in k, and by ν(n) = |Un(k)|, the order of the group Un(k). Of course, ν(n) is a divisor of n; if ν(n) = n, then any generator of Un(k) is called a primitive n-th root of unity [74]. The group Un(k) depends heavily on the base field k. For instance, Upt (k) = {1}, for any positive integer t, if k is a field of characteristic p > 0. If Char(k) = p and p|n then there is no primitive n-th root of unity in k [74, Pag. 295]. k k Furthermore, if k is a finite field with p elements, then ν(n) = gcd (n, p − 1). If Char(k) - n and k is algebraically closed then ν(n) = n. Unless specified otherwise, all algebras, coalgebras, bialgebras, Hopf algebras, tensor products and homomorphisms are over k. For a coalgebra C, we use Sweedler’s Σ-notation: ∆(c) = c(1) ⊗ c(2), (I ⊗ ∆)∆(c) = c(1) ⊗ c(2) ⊗ c(3), etc (summation understood). Let A and H be two Hopf algebras. H is called a right A-module coalgebra if H is a coalgebra in the monoidal category MA of right A-modules, that is there exists / : H ⊗ A → H a morphism of coalgebras such that (H,/) is a right A-module. A morphism between two right A-module coalgebras (H,/) and (H0,/0) is a morphism of coalgebras ψ : H → H0 that is also right A-linear. Fur- thermore, ψ is called unitary if ψ(1H ) = 1H0 . Similarly, A is a left H-module coalgebra if A is a coalgebra in the monoidal category of left H-modules, that is there exists . : H ⊗ A → A a morphism of coalgebras such that (A, .) is also a left H-module.

135 CHAPTER 3. CLASSIFYING BICROSSED PRODUCTS OF QUANTUM GROUPS

Forms and classical descent theory for Hopf algebras

The existence and the classification of forms on a given object (module, algebra, coalgebra, Hopf algebra, etc.) are two challenging problems in descent theory [83]. We shall recall the special case of A-forms of a Hopf algebra H as defined by Parker in [121]. This notion goes back to Pareigis [117], where it was formulated for group algebras. Let k be a field, A a k-algebra and H a Hopf algebra over k. An A-form of H is a Hopf algebra H0 such that there exists a left A-linear isomorphism A ⊗ H0 =∼ A ⊗ H. The following is [121, Question 1.1] where we added the classification part from the classical descent theory [83]: Let A be a k-algebra and H a Hopf algebra. Describe and classify all A-forms of H. The question can be viewed as a special case of the descent type problem for a given functor. Let Hopf be the category of Hopf algebras, AM the category of left A-modules and consider the induction functor F = A ⊗ − : Hopf → AM. The descent theory for the functor F = A ⊗ − refers to describing the image of this functor. In particular, the above question can be restated as a problem which asks for the description of a system of representative in each non-empty fiber F −1(H). There is a relaxed version of the above question formulated first in [71]: Let H be a given Hopf algebra. Describe and classify all Hopf algebras H0 which are A-forms of H, for some k-algebra A. There exists a rich theory related to classical forms of Hopf algebras connected to Hopf Galois extensions and cohomology. For instance, a nice result is [121, Theorem 4.1]: let L/k be a W ∗-Galois field extension for a finite-dimensional, semisimple Hopf algebra W . Any L-form of H is obtained as a certain deformation of H via the so-called commuting action of W on L ⊗ H. For more details on forms for Hopf algebras we refer to [71], [117], [121], [130]. A Hopf algebra E factorizes through two Hopf algebras A and H if there exists injective Hopf algebra maps i : A → E and j : H → E such that the map

A ⊗ H → E, a ⊗ h 7→ i(a)j(h) is bijective. Using the normalizing condition (2.29) we can easily show that in a bicrossed product A ./ H the following relations hold:

(a ./ 1H ) · (b ./ 1H ) = ab ./ 1H , (a ./ 1H ) · (1A ./ h) = a ./ h

(1A ./ h) · (1A ./ g) = 1A ./ hg, (1A ./ h) · (a ./ 1H ) = h(1) . a(1) ./ h(2) / a(2) for all a, b ∈ A, g, h ∈ H. The next theorem provides necessary and sufﬁcient conditions for A ./ H to be a bicrossed product.

Theorem 3.1.1 Let A, H be two Hopf algebras and / : H ⊗ A → H, . : H ⊗ A → A two morphisms of coalgebras satisfying the normalizing conditions (2.29). The following statements are equivalent: (1) A ./ H is a bicrossed product;

136 3.1. MOTIVATING PROBLEMS

(2) (H,/) is a right A-module coalgebra, (A, .) is a left H-module coalgebra and the following compatibilities hold for any a, b ∈ A, g, h ∈ H. g . (ab) = (g(1) . a(1)) (g(2) / a(2)) . b (gh) / a = g / (h(1) . a(1)) (h(2) / a(2))

g(1) / a(1) ⊗ g(2) . a(2) = g(2) / a(2) ⊗ g(1) . a(1) In this case, A ./ H has an antipode given by the formula:

SA./H (a ./ h) = SH (h(2)) .SA(a(2)) ./ SH (h(1)) /SA(a(1)) (3.1) for all a ∈ A and h ∈ H.

Proof: (2) ⇒ (1) Is just [101, Theorem 7.2.2] or [81, Theorem IX 2.3]. (1) ⇒ (2) Follows as a special case of Theorem 2.5.4 if we take the trivial cocycle f : H ⊗H → A, f(g, h) = εH (g)εH (h).

Thus, in the light of Theorem 3.1.1, a matched pair of Hopf algebras is a system (A, H, ., /), where (H,/) is a right A-module coalgebra, (A, .) is a left H-module coalgebra such that the compatibility conditions (2.29) and (2.30)-(2.32) hold.

Examples 3.1.2 1. Let (A, .) be a left H-module coalgebra and consider H as a right A- module coalgebra via the trivial action, i.e. h / a = εA(a)h. Then (A, H, ., /) is a matched pair of Hopf algebras if and only if (A, .) is also a left H-module algebra and the following compatibility condition holds

g(1) ⊗ g(2) . a = g(2) ⊗ g(1) . a (3.2) for all g ∈ H and a ∈ A. In this case, the associated bicrossed product A ./ H = A#H is the semi-direct (or smash) product of Hopf algebras as deﬁned by Molnar [105] in the cocommutative case, for which the compatibility condition (3.2) holds automatically. Thus, A#H is the k-module A ⊗ H, where the multiplication (2.33) takes the form:

(a#h) · (c#g) := a (h(1) . c)# h(2)g (3.3) for all a, c ∈ A, h, g ∈ H, where we denoted a ⊗ h by a#h. A#H is a Hopf algebra with the coalgebra structure given by the tensor product of coalgebras and the antipode

SA#H (a#h) = SH (h(2)) .SA(a)# SH (h(1)) for all a ∈ A and h ∈ H. Similarly, let (H,/) be a right A-module coalgebra and consider A as left H-module coalgebra via the trivial action, i.e h . a := εH (h)a. Then (A, H, ., /) is a matched pair of Hopf algebras if and only if (H,/) is also a right A-module algebra and

g / a(1) ⊗ a(2) = g / a(2) ⊗ a(1) (3.4)

137 CHAPTER 3. CLASSIFYING BICROSSED PRODUCTS OF QUANTUM GROUPS for all g ∈ H and a ∈ A. In this case, the associated bicrossed product A ./ H = A#rH is the right version of the smash product of bialgebras. Thus, A#rH is the k-module A ⊗ H, where the multiplication (2.33) takes the form:

(a#h) · (c#g) := a c(1) #(h / c(2))g (3.5) for all a, c ∈ A, h, g ∈ H, where we denoted a ⊗ h by a#h. A#rH is a Hopf algebra with the coalgebra structure given by the tensor product of coalgebras and the antipode

SA#rH (a#h) = SA(a(2))# SH (h) /SA(a(1)) for all a ∈ A and h ∈ H. 2. Let G and K be two groups, A = k[G] and H = k[K] the group algebras. There exists a bijection between the set of all matched pairs of Hopf algebras (k[G], k[K], ., /) and the set of all matched pairs of groups (G, K, .,˜ /˜) in the sense of Takeuchi [143]. The bijection is given such that there exists an isomorphism of Hopf algebras k[G] ./ k[K] =∼ k[G ./ H], where G ./ H is the Takeuchi’s bicrossed product of groups ([81, pg. 207]. 3. The fundamental example of a bicrossed product is the Drinfel’d double D(H). Let H be a finite dimensional Hopf algebra. Then we have a matched pair of Hopf algebras ((H∗)cop, H, ., /), where the actions / and . are defined by: ∗ ∗ −1 ∗ ∗ −1 h / h := hh ,SH (h(3))h(1)ih(2), h . h := hh ,SH (h(2))? h(1)i (3.6) for all h ∈ H and h∗ ∈ H∗ ([81, Theorem IX.3.5]). The Drinfel’d double of H is the bicrossed product associated to this matched pair, i.e. D(H) = (H∗)cop ./ H. 4. Moreover, Majid generalized the construction of the Drinfel’d double to the infinite dimensional case. Let A and H be two Hopf algebras and λ : H ⊗ A → k a skew pairing. Then there exists a matched pair of Hopf algebras (A, H, . = .λ,/ = /λ), where the actions ., / arise from λ via −1 h / a = h(2)λ (h(1), a(1))λ(h(3), a(2)) = h(2)λ S(h(1))h(3), a (3.7) −1 h . a = a(2)λ (h(1), a(1))λ(h(2), a(3)) = a(2)λ S(h), a(1)S(a(3)) (3.8) for all h ∈ H and a ∈ A ([100, Example 7.2.6]). The corresponding bicrossed product A ./λ H associated to this matched pair is called a generalized quantum double and it will be denoted by Dλ(A, H).

The main theorem which characterizes the bicrossed product is the next theorem due to Majid [101, Theorem 7.2.3]. The normal version of it was recently proven in [44, Proposition 2.2] and more generally in Theorem 2.9.3. First we recall that a Hopf subalgebra A of a Hopf algebra E is called normal if x(1)aS(x(2)) ∈ A and S(x1)ax(2) ∈ A, for all x ∈ E and a ∈ A.

Theorem 3.1.3 Let A, H be two Hopf algebras. A Hopf algebra E factorizes through A and H if and only if there exists a matched pair of Hopf algebras (A, H, ., /) such that E =∼ A ./ H, an isomorphism of Hopf algebras. Furthermore, a Hopf algebra E factorizes through a normal Hopf subalgebra A and a Hopf subalgebra H if and only if E is isomorphic as a Hopf algebra to a semidirect product A# H.

138 3.1. MOTIVATING PROBLEMS

Proof: The bicrossed product A ./ H factorizes through A and H via the canonical maps iA : A → A ./ H, iA(a) = a ./ 1H and iH : H → A ./ H, iH (h) = 1A ⊗ h. Conversely, assume that E factorizes through A and H and we view A and H as Hopf subalgebras of E via the identification A =∼ i(A) and H =∼ j(H). For any a ∈ A and h ∈ H we view ha ∈ E; as E Pt (a,h) (a,h) factorizes through A and H, we can find an unique element i=1 ai ⊗ hi ∈ A ⊗ H such that t X (a,h) (a,h) ha = ai hi i=1 We define the maps / : H ⊗ A → H and . : H ⊗ A → A via:

t t X (a,h) (a,h) X (a,h) (a,h) h / a := εA(ai ) hi , h . a := εH (hi ) ai (3.9) i=1 i=1

Then (A, H, ., /) is a matched pair of Hopf algebras and the multiplication map

A ./ H → E, a ./ h 7→ ah is an isomorphism of Hopf algebras. The details are proven in [101, Theorem 7.2.3]. The ﬁnal statement is proven in [44, Proposition 2.2] or Theorem 2.9.3.

Theorem 3.1.3 proves that the factorization problem for Hopf algebras can be restated in a computational manner: Let A and H be two given Hopf algebras. Describe the set of all matched pairs (A, H, ., /) and classify up to an isomorphism all bicrossed products A ./ H. The Hopf algebras A and H play a symmetric role in a bicrossed product A ./ H. In other words, ﬁnding all matched pairs (H, A, .0,/0) reduces in fact to ﬁnding all matched pairs (A, H, ., /). In the case where both Hopf algebras A and H have bijective antipode we indicate an explicit way of constructing a matched pair (A, H, ., /) out of a given matched pair (H, A, .0,/0) such that the associated Hopf algebras A ./ H and H ./0 A are isomorphic. It is the Hopf algebra version of [11, Proposition 2.5] proved for matched pairs of groups.

Proposition 3.1.4 Let (A, H, ., /) be a matched pair of Hopf algebras with bijective antipodes. We deﬁne the actions .0 : A ⊗ H → H and /0 : A ⊗ H → A given by

0 −1 −1 −1 −1 a . h := SH SH (h(1)) /SA (a(1)) /SA SH (h(2)) .SA (a(2)) 0 −1 −1 −1 −1 a / h := SH SH (h(1)) /SA (a(1)) .SA SH (h(2)) .SA (a(2)) for all a ∈ A and h ∈ H. Then (H, A, .0,/0) is a matched pair of Hopf algebras and there exists an isomorphism of Hopf algebras A ./ H =∼ H ./0 A, where H ./0 A is the bicrossed product associated to (H, A, .0,/0).

139 CHAPTER 3. CLASSIFYING BICROSSED PRODUCTS OF QUANTUM GROUPS

Proof: The proof is a long but straightforward veriﬁcation based on Theorem 3.1.3 as follows: let E := A ./ H be the bicrossed product associated to the matched pair (A, H, ., /). Then E factorizes through A and H; hence E also factorizes through H and A. Next we write down the cross relation in the bicrossed product A ./ H

(1A ./ h) · (a ./ 1H ) = h(1) . a(1) ./ h(2) / a(2)

−1 0 −1 0 0 0 for a = SA (a ) and h = SH (h ), with a ∈ A and h ∈ H and then we apply SA./H to it. We shall obtain an expression for a0 ./ h0 in terms of the actions . and / as follows:

0 0 −1 0 −1 0 −1 0 −1 0 a ./ h = SH SH (h(1)) /SA (a(1)) .SA SH (h(3)) .SA (a(3)) ./ −1 0 −1 0 −1 0 −1 0 ./ SH SH (h(2)) /SA (a(2)) /SA SH (h(4)) .SA (a(4))

Now, the actions .0 and /0 of the new matched pair (H, A, .0,/0) are obtained using (3.9), i.e. we ﬁrst apply εA⊗Id and then Id⊗εH to this formula. Finally, the k-linear map ϕ : A⊗H → H⊗A, 0 ϕ(a ⊗ h) := SH (h) ⊗ SA(a) is bijective and we can easily prove that ψ : A ./ H → H ./ A, ψ := ϕ ◦ SA./H is an isomorphism of Hopf algebras.

140 3.2. THE MORPHISMS BETWEEN TWO BICROSSED PRODUCTS

3.2 The morphisms between two bicrossed products

In order to describe the morphisms between two arbitrary bicrossed products A ./ H and A0 ./0 H0 we need the following:

Lemma 3.2.1 Let C, D and E be three coalgebras. Then there exists a bijective correspondence between the set of all morphisms of coalgebras α : C → D ⊗ E and the set of all pairs (u, p), where u : C → D and p : C → E are morphisms of coalgebras satisfying the symmetry condition

p(c(1)) ⊗ u(c(2)) = p(c(2)) ⊗ u(c(1)) (3.10) for all c ∈ C. Under the above correspondence the morphism of coalgebras α : C → D ⊗ E corresponding to the pair (u, p) is given by:

α(c) = u(c(1)) ⊗ p(c(2)) (3.11) for all c ∈ C.

Proof: Let α : C → D ⊗ E be a morphism of coalgebras. We adopt the temporary notation α(c) = c{0} ⊗ c{1} ∈ D ⊗ E (summation understood). Deﬁne the following two linear maps:

u : C → D, u := (Id ⊗ εE) ◦ α, i.e. u(c) = εE(c{1})c{0}

p : C → E, p := (εD ⊗ Id) ◦ α, i.e. p(c) = εD(c{0})c{1} for all c ∈ C. Then u and p are coalgebra maps as compositions of coalgebra maps. Since α is a coalgebra map we have ∆D⊗E ◦ α(c) = (α ⊗ α) ◦ ∆C (c) for any c ∈ C, which is equivalent to:

c{0}(1) ⊗ c{1}(1) ⊗ c{0}(2) ⊗ c{1}(2) = c(1){0} ⊗ c(1){1} ⊗ c(2){0} ⊗ c(2){1} (3.12)

If we apply Id ⊗ εE ⊗ εD ⊗ Id in (3.12) we obtain c{0} ⊗ c{1} = u(c(1)) ⊗ p(c(2)) that is (3.11) holds. With this form for α and having in mind that u and p are coalgebra maps the equation (3.12) takes the form:

u(c(1)) ⊗ p(c(3)) ⊗ u(c(2)) ⊗ p(c(4)) = u(c(1)) ⊗ p(c(2)) ⊗ u(c(3)) ⊗ p(c(4))

Applying εD ⊗ Id ⊗ Id ⊗ εE to the above identity we obtain the symmetry condition (3.10). Conversely, it is easy to see that any map α given by (3.11), for some coalgebra maps u and p satisfying (3.10), is a coalgebra map and of course the correspondence is bijective.

Now we can describe all Hopf algebra morphisms between two bicrossed products.

Theorem 3.2.2 Let (A, H, ., /) and (A0,H0,.0,/0) be two matched pairs of Hopf algebras. Then there exists a bijective correspondence between the set of all morphisms of Hopf algebras ψ : A ./ H → A0 ./0 H0 and the set of all quadruples (u, p, r, v), where u : A → A0,

141 CHAPTER 3. CLASSIFYING BICROSSED PRODUCTS OF QUANTUM GROUPS p : A → H0, r : H → A0, v : H → H0 are unitary coalgebra maps satisfying the following compatibility conditions:

u(a(1)) ⊗ p(a(2)) = u(a(2)) ⊗ p(a(1)) (3.13)

r(h(1)) ⊗ v(h(2)) = r(h(2)) ⊗ v(h(1)) (3.14) 0 u(ab) = u(a(1)) p(a(2)) . u(b) (3.15) 0 p(ab) = p(a) / u(b(1)) p(b(2)) (3.16) 0 r(hg) = r(h(1)) v(h(2)) . r(g) (3.17) 0 v(hg) = v(h) / r(g(1)) v(g(2)) (3.18) 0 0 r(h(1)) v(h(2)) . u(b) = u(h(1) . b(1)) p(h(2) . b(2)) . r(h(3) / b(3)) (3.19)

0 0 v(h) / u(b(1)) p(b(2)) = p(h(1) . b(1)) / r(h(2) / b(2)) v(h(3) / b(3)) (3.20) for all a, b ∈ A, g, h ∈ H. Under the above correspondence the morphism of Hopf algebras ψ : A ./ H → A0 ./0 H0 corresponding to (u, p, r, v) is given by:

0 0 0 ψ(a ./ h) = u(a(1)) p(a(2)) . r(h(1)) ./ p(a(3)) / r(h(2)) v(h(3)) (3.21) for all a ∈ A and h ∈ H.

Proof: Let ψ : A ./ H → A0 ./0 H0 be a morphism of Hopf algebras. Then

ψ(a ./ h) = ψ((a ./ 1H )(1A ./ h)) = ψ(a ./ 1H )ψ(1A ./ h) for all a ∈ A and h ∈ H. We deﬁne

0 0 0 α : A → A ./ H , α(a) := ψ(a ./ 1H )

0 0 0 β : H → A ./ H , β(h) := ψ(1A ./ h) Then α : A → A0 ⊗ H0 and β : H → A0 ⊗ H0 are morphisms of coalgebras as compositions of coalgebra maps (we recall that the coalgebra structure on A0 ./0 H0 is the tensor product of coalgebras on A0 ⊗ H0). Moreover, α and β are unitary maps and

ψ(a ./ h) = α(a) β(h) (3.22) for all a ∈ A and h ∈ H. It follows from Lemma 3.2.1 applied to α and β that there exist four coalgebra maps u : A → A0, p : A → H0, r : H → A0, v : H → H0 such that

α(a) = u(a(1)) ⊗ p(a(2)), β(h) = r(h(1)) ⊗ v(h(2)) (3.23) and the pairs (u, p) and (r, v) satisfy the symmetry conditions (3.13) and (3.14). Explicitly u, p, r and v are deﬁned by

u(a) = ((Id ⊗ εH0 ) ◦ ψ)(a ./ 1H ), p(a) = ((εA0 ⊗ Id) ◦ ψ)(a ./ 1H )

142 3.2. THE MORPHISMS BETWEEN TWO BICROSSED PRODUCTS

r(h) = ((Id ⊗ εH0 ) ◦ ψ)(1A ./ h), v(h) = ((εA0 ⊗ Id) ◦ ψ)(1A ./ h) for all a ∈ A and h ∈ H. All these maps are unitary coalgebra maps. Now, for any a ∈ A and h ∈ H we have:

ψ(a ./ h) = α(a)β(h) 0 0 = u(a(1)) ./ p(a(2)) · r(h(1)) ./ v(h(2)) 0 0 0 = u(a(1)) p(a(2)) . r(h(1)) ./ p(a(3)) / r(h(2)) v(h(3)) i.e. (3.21) also holds. Thus any bialgebra map ψ : A ./ H → A0 ./0 H0 is uniquely determined by the formula (3.22) for some unitary coalgebra maps α : A → A0 ⊗ H0 and β : H → A0 ⊗ H0 or, equivalently, in a more explicit form as given in (3.21), for some unique quadruple of unitary coalgebra maps (u, p, r, v). Now, a map ψ given by (3.22) is a morphism of algebras if and only if α : A → A0 ./0 H0 and β : H → A0 ./0 H0 are algebra maps and the following commutativity relation holds

β(h) α(b) = α(h(1) . b(1)) β(h(2) / b(2)) (3.24) for all h ∈ H and b ∈ A. Indeed, if ψ is an algebra map then α and β are algebra maps as compositions of algebra maps. On the other hand:

ψ(a ./ h)ψ(b ./ g) = α(a)β(h)α(b)β(g) and ψ (a ./ h)(b ./ g) = α(a)α(h(1) . b(1))β(h(2) / b(2))β(g)

Hence, the relation (3.24) follows by taking a = 1A and g = 1H in the identity above. Con- versely is obvious. Now, we write down the explicit conditions for α and β to be algebra maps. 0 0 0 0 First, we prove that α : A → A ./ H , α(a) = u(a(1)) ./ p(a(2)) is an algebra map if and only if (3.15) and (3.16) hold. Indeed, α(ab) = α(a)α(b) is equivalent to:

0 0 0 0 u(a(1)b(1)) ./ p(a(2)b(2)) = u(a(1)) p(a(2)) . u(b(1)) ./ p(a(3)) / u(b(2)) p(b(3)) (3.25)

If we apply Id ⊗ εH0 to (3.25) we obtain (3.15) while if we apply εA0 ⊗ Id to (3.25) we obtain (3.16). Conversely is obvious. In a similar way we can show that β : H → A0 ./0 H0, β(h) = 0 r(h(1)) ./ v(h(2)) is an algebra map if and only if (3.17) and (3.18) hold. Finally, we prove that the commutativity relation (3.24) is equivalent to (3.19) and (3.20). Indeed, using the expressions of α and β in terms of (u, p) and respectively (r, v), the equation (3.24) is equivalent to:

0 0 0 r(h(1)) v(h(2)) . u(b(1)) ./ v(h(3)) / u(b(2)) p(b(3)) = 0 u(h(1) . b(1)) p(h(2) . b(2)) . r(h(3) / b(3)) ./

0 p(h(4) . b(4)) / r(h(5) / b(5)) v(h(6) / b(6))

If we apply Id ⊗ εH0 to the above identity we obtain (3.19) while if we apply εA0 ⊗ Id to it we get (3.20). Conversely, the commutativity condition (3.24) follows straightforward from (3.19) and (3.20).

143 CHAPTER 3. CLASSIFYING BICROSSED PRODUCTS OF QUANTUM GROUPS

To conclude, we have proved that a bialgebra map ψ : A ./ H → A0 ./0 H0 is uniquely determined by a quadruple (u, p, r, v) of unitary coalgebra maps which satisfy the compatibility conditions (3.13)-(3.20). Moreover, such a bialgebra map ψ : A ./ H → A0 ./0 H0 is given by the formula (3.21). Conversely, the fact that ψ given by (3.21) is a morphism of bialgebras, for some (u, p, r, v) satisfying (3.13)-(3.20) is straightforward and follows directly from the proof. The fact that ψ is an algebra map is proven above. ψ is also a coalgebra map as a composition of coalgebra maps. More precisely, the equation (3.21) can be written in the equivalent form (3.22) namely, ψ = mA0./0H0 ◦ (α ⊗ β), where mA0./0H0 is the multiplication map on the bicrossed product A0 ./0 H0 and α, β are the unitary coalgebra maps obtained from (u, p, r, v) via the formulas (3.23).

In the next corollary A#H will be a semi-direct product of Hopf algebras constructed in (1) of Example 3.1.2 as a special case of a bicrossed product. Thus, A#H is associated to a left H-module structure . on A such that (A, .) is a left H-module coalgebra and a left H-module algebra such that the compatibility condition (3.2) holds.

Corollary 3.2.3 Let A#H and A0#0H0 be two semi-direct products of Hopf algebras associated to two left actions . : H⊗A → and .0 : H0⊗A0 → A0. Then there exists a bijection between the set of all morphisms of Hopf algebras ψ : A#H → A0#0H0 and the set of all quadruples (u, p, r, v), where u : A → A0, r : H → A0 are unitary coalgebra maps, p : A → H0, v : H → H0 are morphism of Hopf algebras satisfying the following compatibility conditions:

u(a(1)) ⊗ p(a(2)) = u(a(2)) ⊗ p(a(1)) (3.26)

r(h(1)) ⊗ v(h(2)) = r(h(2)) ⊗ v(h(1)) (3.27) 0 u(ab) = u(a(1)) p(a(2)) . u(b) (3.28) 0 r(hg) = r(h(1)) v(h(2)) . r(g) (3.29) 0 0 r(h(1)) v(h(2)) . u(b) = u(h(1) . b(1)) p(h(2) . b(2)) . r(h(3)) (3.30)

v(h) p(b) = p(h(1) . b) v(h(2)) (3.31) for all a, b ∈ A, g, h ∈ H. Under the above bijection the morphism of Hopf algebras ψ : A#H → A0#0H0 corresponding to (u, p, r, v) is given by:

0 0 ψ(a#h) = u(a(1)) p(a(2)) . r(h(1)) # p(a(3)) v(h(2)) (3.32) for all a ∈ A and h ∈ H.

Proof: We apply Theorem 3.2.2 in the case that the two right actions /0, / are both the trivial actions. In this case, (3.16) is equivalent to p : A → H0 being an algebra map, i.e. p is a Hopf algebra map while (3.18) is equivalent to v : H → H0 being an algebra map, i.e. v is a Hopf algebra map. Finally, (3.13)-(3.15), (3.17), (3.19) and (3.20) take the simpliﬁed forms (3.26)-(3.28), (3.29), (3.30) and (3.31) respectively.

144 3.2. THE MORPHISMS BETWEEN TWO BICROSSED PRODUCTS

Using Theorem 3.2.2 we can completely describe End(A ./ H), the group of all Hopf algebra endomorphisms of an arbitrary bicrossed product. In particular, we obtain a description of End(D(H)), the group of all Hopf algebra endomorphisms of a Drinfel’d double D(H). Un- fortunately, for an arbitrary H, the results are very technical. For this reason, we will restrict to the case of H = k[G], the group algebra of a ﬁnite group G. The group End(D(k[G])) of all Hopf algebra endomorphisms is described below in full details. We observe that, if H is cocommutative, the right action / of (H∗)cop on H given by (3.6) is the trivial action and hence D(H) = (H∗)cop#H is the semidirect product of (H∗)cop and H associated to the left action . given by (3.6).

Let G be a ﬁnite group. The Drinfel’d double of k[G] is described as follows. Take {eg}g∈G ∗ cop to be a dual basis for the basis {g}g∈G of k[G]. A := (k[G] ) is a Hopf algebra with the multiplication and the comultiplication given by

X X eg · eh := δg,heg, ∆A(eg) := ex ⊗ egx−1 , 1A := eg, εA(eg) := δg,1G x∈G g∈G for all g, h ∈ G, where δ(−, −) is the Kronecker delta. The left action of H = k[G] on A = (k[G]∗)cop given by (3.6) takes the form

−1 g . eh := eghg−1 or (g . f)(z) := f(g zg) (3.33) for all g, h, z ∈ G and f ∈ (k[G]∗)cop. Using the action (3.33), we have that D(k[G]) = ∗ cop (k[G] ) #k[G]. Thus, D(k[G]) is the Hopf algebra having the basis {eh#g}g,h∈G with the multiplication and the comultiplication given by

X (eh # g) · (ex # y) = δh,gxg−1 eh # gy, ∆(eh # g) = (ex # g) ⊗ (ehx−1 # g) x∈G for all h, g, x and y ∈ G. The following gives the parametrization of all Hopf algebra morphisms between two Drin- fel’d doubles associated to two ﬁnite groups G and H. In particular, if G = H, the group EndD(k[G]) of all Hopf algebra endomorphisms is fully described.

Corollary 3.2.4 Let G, H be two ﬁnite groups. Then any Hopf algebra morphism ψ : D(k[G]) → D(k[H]) between the Drinfel’d doubles has the form1

0 X −1 0 0 ψ(eg#g ) = λ(gx , z) ω(g , y) θ(?, x)δ(?, zyz−1) # z v(g ) (3.34) x∈G, y,z∈H for all g, g0 ∈ G, where (λ, ω, θ, v) is a quadruple such that v : G → H is a morphism of groups, θ : H × G → k, λ : G × H → k, ω : G × H → k are three maps satisfying the

1 ∗ We denote by θ(?, y)δ(?, aba−1) ∈ k[H] the k-linear map sending any z ∈ H to θ(z, y)δ{z, aba−1}.

145 CHAPTER 3. CLASSIFYING BICROSSED PRODUCTS OF QUANTUM GROUPS following compatibilities:

θ(1, g) = δg,1G (3.35) X θ(h, x) = 1 (3.36) x∈G X θ(hh0, g) = θ(h, x) θ(h0, x−1g) (3.37) x∈G ω(1, g) = 1 (3.38) ω(g, hh0) = ω(g, h) ω(g, h0) (3.39) X λ(g, y) = δg,1 (3.40) y∈H X λ(x, h) = δ1,h (3.41) x∈G X 0 −1 λ(g, y) λ(g , y h) = δg,g0 λ(g, h) (3.42) y∈H X −1 0 λ(x, h) λ(gx , h ) = δh,h0 λ(g, h) (3.43) x∈G X X θ(h, x) λ(gx−1, h0) = θ(h, x) λ(x−1g, h0) (3.44) x∈G x∈G X −1 −1 0 λ(gx , y) θ(h, x) θ(y hy, g ) = δg,g0 θ(h, g) (3.45) x∈G, y∈H ω(g, h)ωg0, v(g)−1hv(g) = ω(gg0, h) (3.46) X θ(h, xg−1) ω(g, y−1hy)λ(gg0x−1, y) = ω(g0, h) θv(g)−1hv(g), g0 (3.47) x∈G, y∈H λgg0g−1, v(g) h v(g)−1 = λ(g0, h) (3.48) for all g, g0 ∈ G, h, h0 ∈ H. The correspondence ψ ↔ (λ, ω, θ, v) between the set of all Hopf algebra morphisms ψ : D(k[G]) → D(k[H]) and the set of all maps (λ, ω, θ, v) satisfying the compatibility conditions (3.35)-(3.48) is bijective.

Proof: It follows by a direct computation from Corollary 3.2.3 applied for A = (k[G]∗)cop, H = k[H] and the left action . given by (3.33). We shall indicate only a sketch of the proof, the details being left to the reader. First we should notice that any Hopf algebra map v : k[G] → k[H] is uniquely determined by a morphism of the groups which will be denoted also by v : G → H. We shall prove now that any unitary coalgebra map u :(k[G]∗)cop → (k[H]∗)cop is given by the formula u(eg)(h) = θ(h, g) for all g ∈ G, h ∈ H and for a unique map θ : H × G → k satisfying (3.35)-(3.37). Indeed, ﬁrst we notice that a unitary coalgebra map u :(k[G]∗)cop → (k[H]∗)cop is in fact the same

146 3.2. THE MORPHISMS BETWEEN TWO BICROSSED PRODUCTS as a unitary coalgebra map u : k[G]∗ → k[H]∗. From the duality algebras/coalgebras any such map u is uniquely implemented by a map of augmented algebras f : k[H] → k[G] (i.e. f is an ∗ algebra map and εk[G] ◦ f = εk[H]) by the formula u = f , i.e. u(eh) = eh ◦ f, for all h ∈ G. Now, any k-linear map f : k[H] → k[G] is uniquely defined by a map θ : H × G → k as follows: X f(h) = θ(h, x)x x∈G for any g ∈ H. We can easily prove that such a linear map is an endomorphism of k[G] as augmented algebras if and only if the compatibility conditions (3.35)-(3.37) hold. Now, any k-linear map r : k[G] → (k[H]∗)cop is implemented by a unique map ω : G × H → k such that X r(g) = ω(g, y) ey y∈H for all g ∈ G. Such a map r = rω is an unitary morphism of coalgebras if and only if ω satisfies the compatibility conditions (3.38)-(3.39). Finally, any k-linear map p :(k[G]∗)cop → k[H] is given in a unique way by a map λ : G × H → k such that X p(eg) = λ(g, y)y y∈H for all g ∈ G. Such a map p = pλ is a morphism of Hopf algebras if and only if λ satisfies the compatibility conditions (3.40)-(3.43). Hence we have described the set of datum (u, p, r, v) of Corollary 3.2.3. As H = k[H] is cocommutative the compatibility condition (3.27) is trivially fulfilled. Moreover, by a straightforward computation the compatibility conditions (3.26) and (3.28)-(3.31) take the form (3.44)-(3.48). Now, by a direct computation we can show that the expression of the morphism ψ : D(k[G]) → D(k[H]) given by (3.32) takes the following simplified form:

0 X −1 −1 0 0 ψ(eg#g ) = λ(xy , a) λ(gx , c) ω(g , b) θ(?, y)δ(?, aba−1) # c v(g ) a,b,c∈H, x,y∈G X X −1 −1 0 0 = λ(xy , a) λ(gx , c) ω(g , b) θ(?, y)δ(?, aba−1) # c v(g ) a,b,c∈H,y∈G x∈G

(3.43) X −1 0 0 = δa,cλ(gy , c) ω(g , b) θ(?, y)δ(?, aba−1) # c v(g ) a,b,c∈H,y∈G X −1 0 0 = λ(gy , a) ω(g , b) θ(?, y)δ(?, aba−1) # a v(g ) a,b∈H,y∈G i.e., by interchanging the summation indices, we proved that (3.34) holds.

147 CHAPTER 3. CLASSIFYING BICROSSED PRODUCTS OF QUANTUM GROUPS

3.3 The classiﬁcation of bicrossed products

Theorem 3.2.2 can be used to indicate when two arbitrary bicrossed products A ./ H and A0 ./0 H0 are isomorphic. However, since the result is very technical and not so transparent, we restrict ourselves to a special kind of classification, namely the one that stabilizes one of the terms of the bicrossed product. The classification of bicrossed products up to an isomorphism that stabilizes one of the terms has at least two motivations: the first one is the cohomologically point of view which descends to the classification theory of Holders’s groups extensions [133, Theorem 7.34] and the second one is the problem of describing and classifying the A-forms of a Hopf algebra from descent theory which will be detailed in Section 3.4. However, for the examples in Section 3.5, Theorem 3.2.2 will be used in its full generality. Let A and H be two Hopf algebras. We shall classify up to an isomorphism that stabilizes A the set of all Hopf algebras E that factorize through A and H. It follows from Theorem 3.1.3 that any Hopf algebra E that factorizes through A and H is isomorphic to a bicrossed product A ./ H and hence we have to classify all bicrossed products A ./ H associated to all possible matched pairs of Hopf algebras (A, H, ., /). An important concept in this chapter is that of coalgebra lazy 1-cocycle introduced in Defini- 1 tion 2.6.3. We denote by Hl,c(H,A) the set of all coalgebra lazy 1-cocycles of H with coefficients in A. We list below some straightforward properties of this concept that will be used in the sequel.

1 Remark 3.3.1 As we already proved in Section 2.4, Hl,c(H,A) is a group with respect to the 2 convolution product. Furthermore, if r : H → A is a coalgebra lazy 1-cocycle then SA ◦ r = r. 1 −1 Indeed, since r is a coalgebra map, the inverse of r in the group Hl,c(H,A) is r = SA ◦ r, which is still a coalgebra lazy 1-cocycle, i.e. in particular a coalgebra map. Hence, the inverse −1 1 2 2 of SA ◦ r = r in the group Hl,c(H,A) is SA ◦ (SA ◦ r) = SA ◦ r. Thus SA ◦ r = r.

1 Examples 3.3.2 1. If H is cocommutative, then the group Hl,c(H,A) coincides with the group of all unitary coalgebra maps r : H → A with the convolution product. In particular, let G and G0 be two groups, H = k[G] and A = k[G0] the corresponding group 1 0 algebras. Then the group Hl,c(k[G], k[G ]) is isomorphic to the group of all unitary maps r : G → G0.

2. Let H = A := H4 be the Sweedler’s four dimensional Hopf algebra. Then, by a routine 1 computation (see (1) of Lemma 3.5.6) we can prove that Hl,c(H4,H4) is the trivial group with only one element, namely the trivial coalgebra lazy 1-cocycle r : H4 → H4, r(h) = ε(h)1H , for all h ∈ H4.

3. Consider now A = H4 and H = k[Cn]. Let r : k[Cn] → H4 be a unitary coalgebra map. Then (2.62) is trivially fulﬁlled as H is cocomutative. Moreover, since r is a coalgebra map we i 1 have r(c ) ∈ {1, g} for all i ∈ {1, 2, ..., n − 1} and r(1) = 1. Thus Hl,c(k[Cn],H4) is the n−1 abelian group C2 × C2 × · · · × C2 of order 2 . 1 On the other hand, we can prove that Hl,c(H4, k[Cn]) is the group with only one element, namely the trivial coalgebra lazy 1-cocycle r : H4 → k[Cn], r(h) = ε(h)1Cn , for all h ∈ H4.

148 3.3. THE CLASSIFICATION OF BICROSSED PRODUCTS

4. A general method of constructing coalgebra lazy 1-cocycles is the following. Let ψ : A⊗H → A ⊗ H be a left A-linear Hopf algebra isomorphism. Then

r = rψ : H → A, r(h) = ((Id ⊗ εH ) ◦ ψ)(1A ./ h) for all h ∈ H is a coalgebra lazy 1-cocycle. Furthermore, a rψ(h) = rψ(h)a, for all h ∈ H and a ∈ A. Conversely, if r : H → A is a coalgebra lazy 1-cocycle such that Im(r) ⊆ Z(A), then

ψ = ψr : A ⊗ H → A ⊗ H, ψ(a ⊗ h) := a r(h(1)) ⊗ h(2) is a left A-linear Hopf algebra isomorphism. For further details we refer to Corollary 3.3.10.

Deﬁnition 3.3.3 Let A be a Hopf algebra, A ./ H and A ./0 H0 two bicrossed products associated to two matched pairs of Hopf algebras (A, H, ., /) and (A, H0,.0,/0). We say that a morphism of Hopf algebras ψ : A ./ H → A ./0 H0 stabilizes A if the following diagram

i A −−−−→A A ./ H     (3.49) yIdA yψ i A −−−−→A A ./0 H0 is commutative.

A morphism of Hopf algebras ψ : A ./ H → A ./0 H0 stabilizes A if and only if ψ is a morphism of Hopf algebras and left A-modules, where a bicrossed product A ./ H is viewed as a left A-module via the restriction of scalars through the canonical inclusion iA : A → A ./ H. Such (iso)morphisms are fully described in the following:

Theorem 3.3.4 Let A be a Hopf algebra, (A, H, ., /) and (A, H0,.0,/0) two matched pairs of Hopf algebras. Then: (1) There exists a one-to-one correspondence between the set of all Hopf algebra morphisms ψ : A ./ H → A ./0 H0 that stabilizes A and the set of all pairs (r, v), where r : H → A, v : H → H0 are unitary coalgebra maps satisfying the following compatibility conditions for any a ∈ A, g, h ∈ H:

r(h(1)) ⊗ v(h(2)) = r(h(2)) ⊗ v(h(1)) (3.50) 0 r(hg) = r(h(1)) v(h(2)) . r(g) (3.51) 0 v(hg) = v(h) / r(g(1)) v(g(2)) (3.52) 0 h . a = r(h(1)) v(h(2)) . a(1) (SA ◦ r)(h(3) / a(2)) (3.53) v(h / a) = v(h) /0 a (3.54)

Under the above bijection the morphism ψ : A ./ H → A ./0 H0 corresponding to (r, v) is given by: 0 ψ(a ./ h) = a r(h(1)) ./ v(h(2)) (3.55)

149 CHAPTER 3. CLASSIFYING BICROSSED PRODUCTS OF QUANTUM GROUPS for all a ∈ A and h ∈ H. (2) The left A-linear Hopf algebra morphism ψ : A ./ H → A ./0 H0 given by (3.55) is an isomorphism if and only if v : H → H0 is bijective.2

Proof: (1) We shall apply Theorem 3.2.2 for A0 = A. Any morphism of Hopf algebras ψ : A ./ H → A ./0 H0 is given by (3.21) for some unitary coalgebra maps (u, p, r, v) satisfying (3.13)- (3.20). Now, such a morphism ψ : A ./ H → A ./0 H0 makes the diagram (3.49) commutative if and only if the map α : A → A ⊗ H0 constructed in the proof of Theorem 3.2.2 takes the form 0 α(a) = a ⊗ 1H0 . This is equivalent to the fact that u : A → A and p : A → H , constructed in the same proof, are precisely the following: u(a) = a and p(a) = εA(a)1H0 , for any a ∈ A. With these maps u and p, the compatibility relations (3.13)-(3.20) are reduced to (3.50)-(3.54). For instance, (3.19) takes the form

0 (h(1) . a(1))r(h(2) / a(2)) = r(h(1)) v(h(2)) . a which is equivalent to (3.53), as r is invertible in the convolution algebra Hom(H,A) with the inverse SA ◦ r. We also note that (3.20) takes the easier form given by (3.54) which means precisely the fact that the unitary coalgebra map v : H → H0 is also a morphism of right A-modules. Finally, the formula of ψ given by (3.21) takes the simpliﬁed form (3.55). (2) Assume ﬁrst that v : H → H0 is bijective with the inverse v−1. Applying Id ⊗ v−1 in (3.50) we obtain that

r(h(1)) ⊗ h(2) = r(h(2)) ⊗ h(1) for all h ∈ H. Thus, r is a coalgebra lazy 1-cocycle. In particular, it follows from Remark 3.3.1 2 that SA ◦ r = r. Using this observation we can easily prove by a long but straightforward computation that the map deﬁned by

−1 0 0 −1 0 0 −1 0 −1 0 ψ : A ./ H → A ./ H, ψ (a ./ h ) = a SA ◦ r ◦ v (h(1)) ./ v (h(2)) for all a ∈ A and h0 ∈ H0 is the inverse of ψ. Conversely, assume that ψ : A ./ H → A ./0 H0 given by (3.55) is an isomorphism of Hopf algebras and left A-modules. Then, there exists a left A-module Hopf algebra morphism 0 0 ϕ : A ./ H → A ./ H such that ψ ◦ ϕ = IdA./0H0 and ϕ ◦ ψ = IdA./H . It follows from the ﬁrst part of the theorem that there exist two unitary coalgebra maps q : H0 → A and t : H0 → H satisfying the compatibility conditions (3.50)-(3.54), where the pair (., /) is interchanged with (.0,/0), such that ϕ is given by

0 0 0 0 ϕ (a ./ h ) = a q(h(1)) ./ t(h(2))

0 0 for all a ∈ A and h ∈ H . Let h ∈ H. From (ϕ ◦ ψ)(1A ./ h) = 1A ./ h we obtain

1A ./ h = r(h(1))(q ◦ v)(h(2)) ./ (t ◦ v)(h(3))

2That is v :(H,/) → (H0,/0) is an unitary isomorphism of right A-module coalgebras.

150 3.3. THE CLASSIFICATION OF BICROSSED PRODUCTS

If we apply εA on the ﬁrst position in the above equality we obtain t ◦ v = IdH . On the other 0 0 0 0 0 0 hand, let h ∈ H . From (ψ ◦ ϕ)(1A ./ h ) = 1A ./ h we obtain

0 0 0 0 0 0 1A ./ h = q(h(1))(r ◦ t)(h(2)) ./ (v ◦ t)(h(3))

If we apply εA on the ﬁrst position we obtain v ◦ t = IdH0 . Hence, it follows that v is bijective −1 and v = t.

Now, we shall ﬁx two Hopf algebras A and H. We deﬁne the small category MP(A, H) of all matched pairs as follows: the objects of MP(A, H) are the set of all pairs (/, .), such that (A, H, ., /) is a matched pair of Hopf algebras. A morphism ψ :(/, .) → (/0,.0) in the category MP(A, H) is a Hopf algebra map ψ : A ./ H → A ./0 H0 that stabilizes A. In order to classify up to an isomorphism that stabilizes A all Hopf algebras E that factorize through A and H we have to describe the skeleton of the category MP(A, H). This will be done next.

Deﬁnition 3.3.5 Let A and H be two Hopf algebras. Two objects (., /) and (.0,/0) of the category MP(A, H) are called cohomologous and we denote this by (., /) ≈ (.0,/0) if there exists a pair of maps (r, v) such that: (1) r : H → A is a coalgebra lazy 1-cocycle, v : H → H is an unitary isomorphism of coalgebras satisfying the following compatibilities for any g, h ∈ H:

0 r(hg) = r(h(1)) v(h(2)) . r(g) (3.56) 0 v(hg) = v(h) / r(g(1)) v(g(2)) (3.57)

(2) The actions (., /) are implemented from (.0,/0) via (r, v) as follows:

h / a = v−1v(h) /0 a (3.58) 0 −1 0 h . a = r(h(1)) v(h(2)) . a(1) SA ◦ r ◦ v (v(h(3)) / a(2)) (3.59) for all a ∈ A and h ∈ H, where v−1 is the usual inverse of the bijective map v.

Remark 3.3.6 The condition (3.58) is equivalent to saying that v :(H,/) → (H,/0) is an isomorphism of right A-module coalgebras. There exists a trivial object in MP(A, H), namely (/0,.0), where /0, .0 are both the trivial actions. An object (., /) of MP(A, H) is called a coboundary if (., /) is cohomologous with the trivial object (/0,.0). Thus, if we write down the conditions from Deﬁnition 3.3.5 we can easily prove that an object (., /) of MP(A, H) is a coboundary if and only if the right action / is the trivial action and the left action . is implemented by h . a = r(h(1)) a SA r(h(2)) for some coalgebra lazy 1-cocycle r : H → A that is also a morphism of Hopf algebras. This is in fact the necessary and sufﬁcient condition for a bicrossed product A ./ H to be isomorphic as left A-modules and Hopf algebras to a tensor product A ⊗ H.

151 CHAPTER 3. CLASSIFYING BICROSSED PRODUCTS OF QUANTUM GROUPS

The classiﬁcation theorem now follows: the set of all isomorphism types of bicrossed products A ./ H which stabilize A (i.e. the skeleton of the category MP(A, H)) is in bijection with a cohomologically type pointed set H2(A, H) which is in our context the counterpart of the second cohomology group for the classiﬁcation of extensions of an abelian group by a group [133, Theorem 7.34].

Theorem 3.3.7 (The classiﬁcation of bicrossed products) Let A and H be two Hopf algebras. Then ≈ is an equivalence relation on the set MP(A, H) and there exists an one-to-one correspondence between the set of objects of the skeleton of the category MP(A, H)) and the pointed quotient set H2(A, H) := MP(A, H)/ ≈.

Proof: It follows from Theorem 3.3.4 that (., /) ≈ (.0,/0) if and only if there exists a left A- linear Hopf algebra isomorphism ψ : A ./ H → A ./0 H, where A ./ H and A ./0 H are the bicrossed products associated to the matched pairs (A, H, ., /) and respectively (A, H, .0,/0). The compatibility condition (3.59) is exactly (3.53) taking into account (3.58). Thus, ≈ is an equivalence relation on the set MP(A, H) and we are done.

Theorem 3.3.4 has several applications. Three of them are given below. First we shall apply it 0 for inﬁnite dimensional quantum doubles. Let Dλ(A, H) and Dλ0 (A, H ) be two generalized quantum doubles associated to two skew pairings λ, λ0 as constructed in (4) Example 3.1.2. We ∼ 0 shall prove a necessary and sufﬁcient condition for Dλ(A, H) = Dλ0 (A, H ), an isomorphism of Hopf algebras and left A-modules.

Corollary 3.3.8 Let λ : H ⊗ A → k, λ0 : H0 ⊗ A → k be two skew pairings of Hopf algebras, 0 Dλ(A, H) and Dλ0 (A, H ) the generalized quantum doubles. The following are equivalent: ∼ 0 (1) There exists a left A-linear Hopf algebra isomorphism Dλ(A, H) = Dλ0 (A, H ); (2) There exists a pair of maps (r, v), where r : H → A is a coalgebra lazy 1-cocycle, v : H → H0 is an unitary isomorphism of coalgebras satisfying the following four compatibility conditions:

0 r(hg) = r(h(1)) r(g(2)) λ v(h(2)), r(g(3)) SA(r(g(1))) 0 v(hg) = v(h(2))v(g(2))λ SH0 (v(h(1))) v(h(3)), r(g(1)) 0 v(h(2)) λ SH (h(1))h(3), a = v(h(2)) λ SH0 (v(h(1))) v(h(3)), a a(2)r(h(2)) λ SH (h(1)), a(1) λ h(3), a(3) = 0 0 = r(h(1))a(2) λ v(h(3)), a(3) λ SH0 (v(h(2))), a(1) for all a ∈ A, g, h ∈ H.

Proof: First we note that Dλ(A, H) = A ./λ H, where the matched pair (A, H, /λ,.λ) is given in (3.7) and (3.8). Now, we are in position to apply Theorem 3.3.4. The four compatibility conditions from the second statement are precisely (3.51)-(3.54) applied to the matched pairs associated to λ and λ0. In order to simplify these compatibilities, the axioms of the skew pairings

152 3.3. THE CLASSIFICATION OF BICROSSED PRODUCTS

0 λ and λ are used as well as the fact that λ(h, a) = λ(SH (h),SA(a)), for all h ∈ H and a ∈ A [63, Lemma 1.4]. The third compatibility above is (3.54), which means exactly the fact that 0 v : H → H is also a right A-module map.

The next corollary provides necessary and sufﬁcient conditions for a bicrossed product A ./ H to be isomorphic to a smash product A#0H0 such that the isomorphism stabilizes A.

Corollary 3.3.9 Let (A, H, ., /) be a matched pair of Hopf algebras, H0 a Hopf algebra and (A, .0) a left H0-module algebra and coalgebra satisfying the compatibility condition (3.2). The following are equivalent: (1) There exists a left A-linear Hopf algebra isomorphism A ./ H =∼ A#0H0; (2) The right action / is the trivial action and there exists a pair (r, v), where r : H → A is a coalgebra lazy 1-cocycle, v : H → H0 is an isomorphism of Hopf algebras such that the left action . is implemented by the formula: 0 h . a = r(h(1)) v(h(2)) . a (SA ◦ r)(h(3)) (3.60) and the following compatibility condition holds: 0 r(hg) = r(h(1)) v(h(2)) . r(g) (3.61) for all h, g ∈ H and a ∈ A.

Proof: We apply Theorem 3.3.4 for the matched pair (A, H0,.0,/0), where /0 is the trivial action. Using the fact that v is bijective, we obtain from the compatibility condition (3.54) that the right action / is also the trivial action. On the other hand (3.53) takes the equivalent form (3.60) using that r is invertible in the convolution with the inverse SA ◦ r.

As a special case of Theorem 3.3.4 we have the following interesting result:

Corollary 3.3.10 Let A, H, H0 be three Hopf algebras. Then there exists a bijection between the set of all left A-linear Hopf algebra isomorphisms ψ : A ⊗ H → A ⊗ H0 and the set of all pairs (r, v), where v : H → H0 is an isomorphism of Hopf algebras, r : H → A is a coalgebra lazy 1-cocycle and a morphism of Hopf algebras with Im(r) ⊂ Z(A), the center of A. Under the above bijection the left A-linear Hopf algebra isomorphism ψ : A ⊗ H → A ⊗ H0 corresponding to (r, v) is given by:

ψ(a ⊗ h) = a r(h(1)) ⊗ v(h(2)) (3.62) for all a ∈ A and h ∈ H.

Proof: We consider /, /0, . and .0 to be all the trivial actions in Theorem 3.3.4. The compatibility conditions (3.51)-(3.53) implies in this case that r and v are also algebra maps, while (3.53) becomes r(h(1)) a (SA ◦ r)(h(2)) = εH (h)a for all a ∈ A, h ∈ H which is equivalent to the centralizing condition r(h)a = ar(h), for all a ∈ A and h ∈ H.

153 CHAPTER 3. CLASSIFYING BICROSSED PRODUCTS OF QUANTUM GROUPS

Corollary 3.3.11 Let A, H and H0 be three Hopf algebras. Then H and H0 are isomorphic as Hopf algebras if and only if there exists a left A-linear Hopf algebra isomorphism A ⊗ H =∼ A ⊗ H0.

0 0 Proof: If f : H → H is an isomorphism of Hopf algebras, then IdA ⊗ f : A ⊗ H → A ⊗ H is a left A-linear Hopf algebra isomorphism. Conversely, if ψ : A ⊗ H → A ⊗ H0 is a left A-linear Hopf algebra isomorphism then the map v : H → H0 constructed in the proof of Corollary 3.3.10 is in fact an isomorphism of Hopf algebras.

3.4 Bicrossed descent theory and deformations of Hopf algebras

This section is devoted to the converse of the factorization problem. Let A ⊆ E be a given extension of Hopf algebras. We are going to study the set of all Hopf subalgebras H of E such that E factorizes through A and H. This is a descent type question and has two subsequent problems which we formulate below. Our main result of this section states that the factorization index of a given extension A ⊆ E can be computed by the formula [E : A]f = |HA2(H,A | (., /))|. Our notations, such as HA2(H,A | (., /)), are inspired from the cohomology theory of the extension problem. However, at ﬁrst sight there is no connection between our theory and the non-abelian cohomology for Hopf algebras introduced in [112] or [113]. First we need to introduce the following concept:

Deﬁnition 3.4.1 Let A ⊆ E be an extension of Hopf algebras. A factorization A-form of E is a Hopf subalgebra H ⊆ E such that E factorizes through A and H.

We denote by F(A, E) the small category, possibly empty, of all factorization A-forms of E in which morphisms between two objects H and H0 are Hopf algebra maps. It follows from Theorem 3.1.3 that a factorization A-form of E can be defined equivalently as a triple (H, /, .) consisting of a Hopf subalgebra H of E and two linear maps / : H ⊗ A → H, . : H ⊗ A → A such that (A, H, /, .) is a matched pair of Hopf algebras and the multiplication map · : A ./ H → E, a ./ h 7→ ah is a Hopf algebra isomorphism. But there is more: the multiplication map · : A ./ H → E, a ./ h 7→ ah is also left a A-module map and this elementary observation will be important below. With this concept we can introduce a descent type theory for Hopf algebra extensions which consists of the following two problems. Let A ⊆ E be a given extension of Hopf algebras. Existence of forms: Does there exist a factorization A-form of E, i.e. is the set F(A, E) nonempty? Description and classification of forms: If a factorization A-form of E exists, describe and classify up to isomorphism all factorization A-forms of E. The problem of existence of forms has to be treated ”case by case” for every given Hopf algebra extension A ⊆ E, a computational part of it can not be avoided. This was also the approach used in the similar problem at the level of groups, i.e. corresponding to the Hopf algebra extension k[A] ⊆ k[G], for two groups A and G with A ≤ G. The problem of finding all factorizations

154 3.4. DESCENT THEORY AND DEFORMATIONS OF HOPF ALGEBRAS of a given group G was started in the 80’s and there is a very rich literature on the subject: see for instance [11], [12], [31], [68], [88], [89], [126] and the references therein. For example, if E = k[A6] and A is a proper Hopf subalgebra, then F(A, k[A6]) is the empty set. This is based 3 on the fact that the alternating group A6 has no proper factorizations [146]. In what follows we give the complete answer to the second problem concerning the description and classification of all forms. Let H ∈ F(A, E) be a given factorization A-form of E and (H, /, .) be the associated matched pair such that A ./ H → E, a ./ h 7→ ah is a left A- modules and Hopf algebra isomorphism. We shall describe all factorization A-forms of E in terms of (H, /, .) and a coalgebra lazy 1-cocycle r : H → A satisfying a certain compatibility condition. More precisely, H0 will appear as a new type of deformation of the Hopf algebra H that will be defined via a coalgebra lazy 1-cocycle r : H → A. The classification of all factorization A-forms of E is also given by proving that the skeleton of the category F(A, E) is in bijection to a cohomological type object that will be denoted by HA2(H,A | (., /)). In order to prove these results we need to introduce a few more concepts:

Deﬁnition 3.4.2 Let A ⊆ E be an extension of Hopf algebras. We deﬁne the factorization index of the extension E/A as the cardinal of the set of types of isomorphisms of all factorization A-forms of E and will be denoted by [E : A]f . Hence

[E : A]f = | F sk(A, E) | where, F sk(A, E) is the skeleton of the category F(A, E). The extension A ⊆ E is called rigid (resp. of ﬁnite descent type) if [E : A]f = 1 (resp. [E : A]f is a ﬁnite number). We shall write [E : A]f = 0, if F(A, E) is empty.

We write down explicitly what a rigid extension of Hopf algebras E/A means: [E : A]f = 1 if and only if any two factorization A-forms H and H0 of E are isomorphic as Hopf algebras. Explicitly, this means the following: if E =∼ A ./ H =∼ A ./0 H0, then H =∼ H0. This is a Krull-Schmidt-Azumaya type theorem for bicrossed products of two Hopf algebras. Thus rigid extensions of Hopf algebras are exactly those for which the decomposition as a bicrossed product is unique.

Examples 3.4.3 1. Several examples of rigid Hopf algebra extensions are given in Section 3.5. In fact, most of the Hopf algebra extensions E/A have the factorization index [E : A]f equal to 0 or 1. Examples of extensions E/A for which [E : A]f ≥ 2 are quite rare, which makes them easy to identify. We provide below three examples of such extensions.

2. Let Sn be the symmetric group on n letters. As usually, we view S3 as a subgroup of S4 by letting 4 to be a ﬁxed point. Then the extension k[S3] ⊆ k[S4] has factorization index 2. More details can be found in Example 3.6.2. 3. The following is an example of a Hopf algebra extension of factorization index 3 arising also + from exact factorizations of groups. Let A = A9 and G be the group of Lie type G = Ω8 (2). It 3For an elementary proof of this fact see the ﬁrst version of [11, Proposition 3.12], available only on arXiv at http://arxiv.org/pdf/math/0703471v1.pdf.

155 CHAPTER 3. CLASSIFYING BICROSSED PRODUCTS OF QUANTUM GROUPS was proven in [89, Chapter 16, pg.90], using Magma, that G has precisely three non-isomorphic + exact factorizations. Thus the extension of Hopf algebras k[A9] ⊆ k[Ω8 (2)] has factorization index 3. 4. Example (2) above can be generalized in order to obtain Hopf algebra extensions of a given ∼ degree. Let n be a positive integer and Sn the symmetric group on n letters and let A = Sn−1 be the stabilizer of 1. Then we have an exact factorization Sn = AG, for a regular subgroup G ≤ Sn of order n. As any group of order n has a regular action on n points, it follows that the factorization index of the extension k[Sn−1] ⊂ k[Sn] equals the number of types of isomorphisms of groups of order n. In particular, k[S7] ⊂ k[S8] has factorization index 5 as, up to an isomorphism, there exists exactly 5 types of groups of order 8.

In order to provide a classiﬁcation of forms we need to introduce two more concepts:

Deﬁnition 3.4.4 Let (A, H, ., /) be a matched pair of Hopf algebras. A coalgebra lazy 1- 1 cocycle r ∈ Hl,c(H,A) is called a (., /)-cocycle if the following compatibility holds: r h / r(g(1)) g(2) = r(h(1)) h(2) . r(g) (3.63) for all g, h ∈ H.

1 1 Let HA (H,A | (., /)) ⊆ Hl,c(H,A) be the set of all (., /)-cocycles. The trivial cocyle r : 1 H → A, r(h) = ε(h)1A is a (., /)-cocycle but in general HA (H,A | (., /)) is not a subgroup 1 of Hl,c(H,A). We shall introduce now the following:

Deﬁnition 3.4.5 Let (A, H, ., /) be a matched pair of Hopf algebras. Two (., /)-cocycles r and R are called equivalent and we denote this by r ∼ R if there exists σ : H → H an unitary automorphism of the coalgebra H such that σ (h / r(g(1))) g(2) = σ(h) /R(σ(g(1))) σ(g(2)) (3.64) for all g, h ∈ H.

As a conclusion of all the results proven below, the main theorem of this section is the following:

Theorem 3.4.6 (Classiﬁcation of forms) Let A ⊆ E be an extension of Hopf algebras, H a factorization A-form of E with the associated matched pair (A, H, ., /). Then: (1) ∼ is an equivalence relation on HA1(H,A | (., /)). We denote by HA2(H,A | (., /)) the quotient set HA1(H,A | (., /))/ ∼. (2) There exists a bijection between the set of types of isomorphisms of all factorization A-forms of E4 and HA2(H,A | (., /)). In particular, the factorization index of the extension A ⊆ E is computed by the formula: [E : A]f = |HA2(H,A | (., /))|

4That is F sk(A, E), the skeleton of the category F(A, E)

156 3.4. DESCENT THEORY AND DEFORMATIONS OF HOPF ALGEBRAS

We prove this result in three steps, each of them of interest in its own right, which we have called: deformation of a Hopf algebra, deformations of forms and ﬁnally the description of forms. This results together with Theorem 3.4.6 shows the following: in order to describe and to classify the set of all factorization A-forms on E it is enough to know only one object in H ∈ F(A, E). First we prove the following theorem where a general deformation of a given Hopf algebra H is proposed. This deformation is associated to an arbitrary matched pair of Hopf algebras (A, H, ., /) and to an (., /)-cocycle r : H → A and it will be used in order to describe all factorization A-forms of E.

Theorem 3.4.7 (Deformation of a Hopf algebra) Let (A, H, ., /) be a matched pair of Hopf algebras and r : H → A an (., /)-cocycle. Let Hr := H, as a coalgebra, with the new multiplication • on H deﬁned for any h, g ∈ H as follows:

h • g := h / r(g(1)) g(2) (3.65)

Then Hr = (Hr, •, 1H , ∆H , εH ) is a Hopf algebra with the antipode given by

S : Hr → Hr,S(h) := SH (h(2)) / (SA ◦ r)(h(1)) (3.66) for all h ∈ H, called the r-deformation of H.

Proof: Using the normalizing conditions (2.29) and the fact that r : H → A is a unitary map, 1H remains the unit for the new multiplication • given by (3.65). On the other hand for any h, g, t ∈ H we have: (h • g) • t = h / r(g(1)) g(2) • t = (h / r(g(1)))g(2) / r(t(1)) t(2) (2.31) = h / r(g(1)) / g(2) . r(t(1)) g(3) / r(t(2)) t(3) = h / r(g(1))(g(2) . r(t(1))) g(3) / r(t(2)) t(3) (3.63) = h / r (g(1) / r(t(1))) t(2) g(2) / r(t(3)) t(4) (2.62) = h / r (g(1) / r(t(1))) t(3) g(2) / r(t(2)) t(4) = h • g / r(t(1)) t(2) = h • (g • t)

Thus, the multiplication • is associative and has 1H as a unit. Next we prove that εH : H → k and ∆H : H → H ⊗ H are algebra maps with respect to the new multiplication (3.65). Indeed, for any h, g ∈ H we have: εH (h • g) = εH h / r(g(1)) g(2) = εH (h)εH (g(1))εH (g(2)) = εH (h)εH (g)

157 CHAPTER 3. CLASSIFYING BICROSSED PRODUCTS OF QUANTUM GROUPS and ∆H (h • g) = ∆H h / r(g(1)) g(2) = h(1) / r(g(1)(1)) g(2)(1) ⊗ h(2) / r(g(1)(2)) g(2)(2) = h(1) / r(g(1)) g(3) ⊗ h(2) / r(g(2)) g(4) (2.62) = h(1) / r(g(1)) g(2) ⊗ h(2) / r(g(3)) g(4)

= h(1) • g(1) ⊗ h(2) • g(2)

Thus Hr = (Hr, •, 1H , ∆H , εH ) is a bialgebra. It remains to prove that S : Hr → Hr given by (3.66) is an antipode for Hr. Indeed, for any h ∈ H we have: S(h(1)) • h(2) = SH (h(2)) /SA ◦ r(h(1)) • h(3) = SH (h(2)) /SA ◦ r(h(1)) / r(h(3)) h(4) (2.62) = SH (h(3)) /SA ◦ r(h(1)) / r(h(2)) h(4) = SH (h(3)) / SA ◦ r(h(1))r(h(2)) h(4)

= SH (h(1)) h(2)

= εH (h)1H for all h ∈ H, where we use the fact that SA ◦ r(h(1)) r(h(2)) = εH (h)1A. Analogous, one can show that h(1) • S(h(2)) = εH (h)1H , for all h ∈ H and the proof is ﬁnished.

Remarks 3.4.8 1. Assume that in Theorem 3.4.7 the coalgebra lazy 1-cocycle r : H → A is the trivial cocycle r(h) = εH (h)1A or the right action / is the trivial action of A on H, i.e. h / a = εA(a)h, for all h ∈ H and a ∈ A. Then Hr = H as Hopf algebras. In general, the new Hopf algebra Hr may not be isomorphic to H as a Hopf algebra. Indeed, the second example of Example 3.4.3 shows that k[C2 × C2] and k[C4] are both factorization k[S3]-forms of k[S4]. Using Theorem 3.4.10 proven below, we obtain that the Hopf algebra k[C2 × C2] is isomorphic to k[C4]r, for some (., /)-cocycle r : k[C4] → k[S3]. However, in Theorem 3.4.9 we r prove that there exists a bicrossed product A ./ Hr which is isomorphic as Hopf algebras and left A-modules to A ./ H. 2. As an historical background, Drinfel’d [66] defined a deformation for the comultiplication of a Hopf algebra H, using an invertible element R ∈ H ⊗ H, called twist, in order to obtain a new Hopf algebra HR. As an algebra HR is equal to H, but the comultiplication is given −1 by the formula ∆R(h) := R∆(h)R , for all h ∈ H. In order for HR to be a Hopf algebra, R has to satisfy some compatibility conditions which are equivalent to saying that R−1 is a cocycle in the Harrison cohomology. Drinfel’d twist deformation proved itself to be crucial in the classification theory of finite dimensional Hopf algebras as well as in the representation theory of fusion categories. The dual case was introduced by Doi [60]: the algebra structure of a Hopf algebra H was deformed using a normalized Sweedler cocycle σ : H ⊗ H → k as follows: let Hσ = H, as a

158 3.4. DESCENT THEORY AND DEFORMATIONS OF HOPF ALGEBRAS coalgebra, with the new multiplication given by

−1 h · g := σ(h(1), g(1)) h(2)g(2) σ (h(3), g(3)) for all h, g ∈ H. Then Hσ is a new Hopf algebra [60, Theorem 1.6] and among several interesting applications in quantum groups we mention that the Drinfel’d double D(H) is a special case of this deformation [63, Remark 2.3].

The next step shows that if H is a given factorization A-form of E, then Hr remains a factorization A-form of E, for any (., /)-cocycle r : H → A.

Theorem 3.4.9 (Deformation of forms) Let (A, H, ., /) be a matched pair of Hopf algebras, r r : H → A an (., /)-cocycle and Hr the r-deformation of H. We deﬁne a new action . : Hr ⊗ A → A given as follows for any h ∈ Hr and a ∈ A:

r h . a := r(h(1)) h(2) . a(1) (SA ◦ r)(h(3) / a(2)) (3.67)

r Then (A, Hr,. ,/) is a matched pair of Hopf algebras and the k-linear map

r r ψ : A ./ Hr → A ./ H, ψ(a ./ h) = a r(h(1)) ./ h(2) (3.68)

r for all a ∈ A and h ∈ H is a left A-linear Hopf algebra isomorphism, where A ./ Hr is the r bicrossed product associated to the matched pair (A, Hr,. ,/).

In particular, Hr is a factorization A-form of A ./ H.

r Proof: Instead of using a straightforward but rather long computation to prove that (A, Hr,. ,/) r is a matched pair we use Theorem 3.1.1. In order to do this we ﬁrst prove that . : Hr ⊗ A → A given by (3.67) is a coalgebra map. Indeed, for any a ∈ A, h ∈ H we have:

r r (h . a)(1) ⊗ (h . a)(2) = r(h(1)) h(3) . a(1) SA ◦ r h(6) / a(4) ⊗ r(h(2)) h(4) . a(2) SA ◦ r h(5) / a(3) (2.62) = r(h(1)) h(2) . a(1) SA ◦ r h(6) / a(4) ⊗ r(h(3)) h(4) . a(2) SA ◦ r h(5) / a(3) (2.62) = r(h(1)) h(2) . a(1) SA ◦ r h(5) / a(3) ⊗ r(h(3)) h(4) . a(2) SA ◦ r h(6) / a(4) (2.32) = r(h(1)) h(2) . a(1) SA ◦ r h(4) / a(2) ⊗ r(h(3)) h(5) . a(3) SA ◦ r h(6) / a(4) (2.62) = r(h(1)) h(2) . a(1) SA ◦ r h(3) / a(2) ⊗ r(h(4)) h(5) . a(3) SA ◦ r h(6) / a(4) r r = h(1) . a(1) ⊗ h(2) . a(2)

159 CHAPTER 3. CLASSIFYING BICROSSED PRODUCTS OF QUANTUM GROUPS

r Thus, . : Hr ⊗ A → A is a coalgebra map and moreover, the normalizing conditions (2.29) r r are trivially fulﬁlled for the pair of actions (. ,/). In order to prove that (A, Hr,. ,/) is a matched pair of Hopf algebras we use Theorem 3.1.1 as follows: ﬁrst, observe that the map ψ : A ⊗ Hr → A ./ H, ψ(a ⊗ h) = a r(h(1)) ./ h(2) is an unitary isomorphism of coalgebras with the inverse given by

−1 −1 ψ : A ./ H → A ⊗ Hr, ψ (a ./ h) = a SA ◦ r (h(1)) ⊗ h(2) for all a ∈ A and h ∈ H. The key step follows by using [12, Lemma 2.6]: there exists a unique algebra structure on the coalgebra A ⊗ Hr such that ψ becomes an isomorphism of Hopf algebras and this is obtained by transferring the algebra structure from the Hopf algebra A ./ H via the isomorphism of coalgebras ψ, i.e. is given by:

(a ⊗ h) (b ⊗ g) := ψ−1ψ(a ⊗ h) ψ(b ⊗ g) for all a, b ∈ A and h, g ∈ Hr = H. If we prove that this algebra structure on the tensor product of coalgebras A ⊗ Hr is exactly the one given by (2.33) associated to the new pair of actions (.r,/) on a bicrossed product then the proof is ﬁnished by using Theorem 3.1.1. Indeed, for any a, b ∈ A and g, h ∈ H we have:

(a ⊗ h) (b ⊗ g) = ψ−1ψ(a ⊗ h) ψ(b ⊗ g) −1 = ψ a r(h(1)) ./ h(2) b r(g(1)) ./ g(2)

−1 = ψ a r(h(1)) h(2) . b(1)r(g(1)) ./ h(3) / b(2)r(g(2)) g(3) = a r(h(1)) h(2) . b(1)r(g(1)) (SA ◦ r) h(3) / b(2)r(g(2)) g(4) ⊗ h(4) / b(3)r(g(3)) g(5) (2.62) = a r(h(1)) h(2) . b(1)r(g(1)) (SA ◦ r) h(3) / b(2)r(g(2)) g(3) ⊗ h(4) / b(3)r(g(4)) g(5) = a r(h(1)) h(2) . b(1)r(g(1)) (SA ◦ r) (h(3) / b(2)) / r(g(2)) g(3) ⊗ h(4) / b(3)r(g(4)) g(5) h i = a r(h(1)) h(2) . b(1)r(g(1)) (SA ◦ r) h(3) / b(2)) /r(g(2)) g(3) ⊗ h(4) / b(3)r(g(4)) g(5) (3.63) h i = a r(h(1)) h(2) . b(1)r(g(1)) SA r h(3) / b(2) h(4) / b(3) .r(g(2)) ⊗ h(5) / b(4)r(g(3)) g(4) (2.30) = a r(h(1)) h(2) . b(1) h(3) / b(2) .r(g(1)) SA h(5) / b(4) .r(g(2)) (SA ◦ r h(4) / b(3) ⊗ h(6) / b(5)r(g(3)) g(4)

160 3.4. DESCENT THEORY AND DEFORMATIONS OF HOPF ALGEBRAS

(2.62) = a r(h(1)) h(2) . b(1) h(3) / b(2) .r(g(1)) SA h(4) / b(3) .r(g(2)) (SA ◦ r) h(5) / b(4) ⊗ h(6) / b(5)r(g(3)) g(4) = a r(h(1)) h(2) . b(1) (SA ◦ r) h(3) / b(2) ⊗ h(4) / b(3)r(g(1)) g(2) = a r(h(1)) h(2) . b(1) (SA ◦ r) h(3) / b(2) ⊗ h(4) / b(3) •g r r r = a (h(1) . b(1)) ⊗ (h(2) / b(2)) • g = (a ./ h)(b ./ g) where • is the multiplication on Hr. The proof is now completely ﬁnished.

The same type of result as in Theorem 3.4.9 was proved for twisted tensor product algebras in [76, Theorem 4.4], where it was called the invariance under twisting theorem. Now, we shall prove the converse of Theorem 3.4.9: if H is a given factorization A-form of E then any other form H is isomorphic as a Hopf algebra with some Hr, where Hr is an r- deformation of H for some (., /)-cocycle r : H → A. Moreover the next theorem can be viewed as a bicrossed version of [121, Theorem 4.1] where a similar result for classical L-forms of a Hopf algebra H is proved. Let L/k be a W ∗-Galois ﬁeld extension for a ﬁnite-dimensional, semisimple Hopf algebra W . Any L-form of H is obtained as a certain deformation of H via the so-called commuting action of W on L ⊗ H.

Theorem 3.4.10 (Description of forms) Let A ⊆ E be an extension of Hopf algebras, H a factorization A-form of E with the associated matched pair (A, H, ., /) and let H be an arbitrary ∼ factorization A-form of E. Then there exists an isomorphism of Hopf algebras H = Hr, for some (., /)-cocycle r : H → A, where Hr is the r-deformation of H.

Proof: Let (A, H, ., /) be the matched pair of Hopf algebras such that the multiplication map 0 0 mE : A ./ H → E is a left A-linear Hopf algebra isomorphism. Let (A, H,. ,/ ) be the 0 matched pair associated to H such that the multiplication map mE : A ./ H → E is an −1 0 isomorphism of Hopf algebras. Then ψ := mE ◦ IdE ◦ mE : A ./ H → A ./ H is a left A- linear Hopf algebra isomorphism as a composition of such maps. Now we apply Theorem 3.3.4: 0 this left A-linear Hopf algebra isomorphism ψ : A ./ H → A ./ H is uniquely determined by a pair (r, v) consisting of a coalgebra lazy 1-cocycle r : H → A and a unitary isomorphism of coalgebras v : H → H satisfying the compatibility conditions (3.51)-(3.54), with (/, .) and (/0,.0) interchanged, that is 0 0 0 0 0 r(h g ) = r(h(1)) v(h(2)) . r(g ) (3.69) 0 0 0 0 0 v(h g ) = v(h ) / r(g(1)) v(g(2)) (3.70) 0 0 0 0 0 0 h . a = r(h(1)) v(h(2)) . a(1) (SA ◦ r)(h(3) / a(2)) (3.71) v(h0 /0 a) = v(h0) / a (3.72)

0 0 0 for all h , g ∈ H and a ∈ A. Moreover, ψ : A ./ H → A ./ H is given by: 0 0 0 0 ψ(a ./ h ) = a r(h(1)) ./ v(h(2))

161 CHAPTER 3. CLASSIFYING BICROSSED PRODUCTS OF QUANTUM GROUPS

0 for all a ∈ A and h ∈ H. We deﬁne now r : H → A, r := r ◦ v−1

If we prove that r is a (., /)-cocycle and v : H → Hr is a Hopf algebra isomorphism the proof is ﬁnished. First, notice that r is a coalgebra lazy 1-cocycle as r, v are both unitary coalgebra maps and r is a coalgebra lazy 1-cocycle. We have to show that the compatibility condition (3.63) holds for r. Indeed, from (3.69) and (3.70) we obtain: −1 0 0 0 0 0 0 r ◦ v [ v(h ) / r(g(1)) v(g(2)) ] = r(h(1)) v(h(2)) . r(g ) (3.73) 0 0 0 −1 for all h , g ∈ H. Let h, g ∈ H and write the compatibility condition (3.73) for h = v (h) and g0 = v−1(g). We obtain r h / r(g(1)) g(2) = r(h(1)) h(2) . r(g) that is (3.63) holds and hence r : H → A is a (., /)-cocycle. Finally, v : H → Hr is a coalgebra isomorphism as the coalgebra structure on Hr coincides with the one on H. Hence, we are left 0 0 to prove that v is also an algebra map. Indeed, for any h , g ∈ H we have:

0 0 (3.70) 0 0 0 (3.65) 0 0 v(h g ) = v(h ) / r(g(1)) v(g(2)) = v(h ) • v(g ) where we denoted by • the multiplication on Hr as deﬁned by (3.65). Hence v : H → Hr is a Hopf algebra isomorphism and the proof is ﬁnished.

Now we are ready to prove Theorem 3.4.6.

The proof of Theorem 3.4.6: It follows from Theorem 3.4.10 that if H is an arbitrary factor- ∼ ization A-form of E, then there exists an isomorphism of Hopf algebras H = Hr, for some (., /)-cocycle r : H → A. Thus, in order to classify all factorization A-forms on E we can consider only r-deformations of H, for various (., /)-cocycles r : H → A.

Now let r, R : H → A be two (., /)-cocycles. As the coalgebra structure on Hr and HR coincide with the one of H, we obtain that the Hopf algebras Hr and HR are isomorphic if and only if there exists σ : H → H a unitary coalgebra isomorphism such that σ : Hr → HR is also an algebra map. Taking into account the definition of the multiplication on Hr given by (3.65) we obtain that σ is an algebra map if and only if the compatibility condition (3.64) of Definition 3.4.5 holds, i.e. r ∼ R. Hence, r ∼ R if and only if σ : Hr → Hr is an isomorphism of Hopf algebras. Thus we obtain that ∼ is an equivalence relation on HA1(H,A | (., /)) and the map 2 sk HA (H,A | (., /)) → F (A, E), r 7→ Hr where r is the equivalence class of r via the relation ∼, is well defined and a bijection between sets. This finishes the proof.

Mutatis-mutandis, Theorem 3.4.6 can be viewed as a bicrossed version for Hopf algebras of the classical result in descent theory: if k ⊆ l is a faithfully ﬂat extension of commutative rings then the Amitsur cohomology group is isomorphic to the relative Picard group Pic(l/k) (see [83]).

162 3.5. EXAMPLES

3.5 Examples

This section is devoted to the construction of some explicit examples: for two given Hopf algebras A and H we will describe and classify all Hopf algebras E that factorize through A and H. Furthermore, for any such Hopf algebra E the factorization index [E : A]f is computed. The are three steps that we have to go through: ﬁrst of all we have to compute the set of all matched pairs between A and H. Then we have to describe by generators and relations all bicrossed products A ./ H associated to these matched pairs. Finally, using the power of Theorem 3.2.2, we shall classify up to an isomorphism these bicrossed products A ./ H. As an application, the group Aut Hopf (A ./ H) of all Hopf algebra automorphisms of a given bicrossed product is computed. For a Hopf algebra H, G(H) is the set of group-like elements of H and for g, h ∈ G(H) we denote by Pg,h(H) the set of all (g, h)-primitive elements, that is

Pg, h(H) = {x ∈ H | ∆H (x) = x ⊗ g + h ⊗ x}

The following result is very useful in computing all matched pairs between A and H:

Lemma 3.5.1 Let (A, H, ., /) be a matched pair of Hopf algebras, a, b ∈ G(A) and g, h ∈ G(H). Then: (1) g . a ∈ G(A) and g / a ∈ G(H);

(2) If x ∈ Pa, b(A), then g / x ∈ Pg/a, g/b(H) and g . x ∈ Pg.a, g.b(A);

(3) If y ∈ Pg, h(H), then y / a ∈ Pg/a, h/a(H) and y . a ∈ Pg.a, h.a(A).

In particular, if x is an (1A, b)-primitive element of A, then g . x is an (1A, g . b)-primitive element of A and g / x is an (g, g / b)-primitive element of H.

Proof: Straightforward: in fact for (1)-(3) we only use that / and . are coalgebra maps, i.e. (2.1) and (2.2) hold. For the ﬁnal statement we use the fact that the normalizing conditions (2.29) hold for (., /).

From now on, Cn will be the cyclic group of order n generated by c and k will be a ﬁeld of characteristic 6= 2. Let A := H4 be the Sweedler’s 4-dimensional Hopf algebra having {1, g, x, gx} as a basis subject to the relations:

g2 = 1, x2 = 0, xg = −gx with the coalgebra structure and antipode given by:

∆(g) = g ⊗ g, ∆(x) = x ⊗ 1 + g ⊗ x, ∆(gx) = gx ⊗ g + 1 ⊗ gx

ε(g) = 1, ε(x) = 0,S(g) = g, S(x) = −gx

In order to compute the set of all matched pairs between H4 and k[Cn] we need the following elementary result.

163 CHAPTER 3. CLASSIFYING BICROSSED PRODUCTS OF QUANTUM GROUPS

Lemma 3.5.2 Let k be a ﬁeld of characteristic 6= 2, H4 the Sweedler’s 4-dimensional Hopf algebra and k[Cn] the group algebra of Cn. Then:

i j Pci, cj (k[Cn]) = {λc − λc | λ ∈ k}

G(H4) = {1, g},P1, 1(H4) = {0},Pg, g(H4) = {0}

P1, g(H4) = {α − αg + β x | α, β ∈ k},Pg, 1(H4) = {α − αg + β gx | α, β ∈ k} for all i, j = 0, 1, ··· , n − 1.

Proof: Everything is just a straightforward computation. For example, let z = λ0 + λ1c + 2 n−1 t ∗ λ2c + ··· + λn−1c ∈ Pci, cj (k[Cn]), for some λj ∈ k. Let t 6= i, t 6= j and apply Id ⊗ (c ) i j t ∗ t in ∆(z) = z ⊗ c + c ⊗ z (where (c )t=0,··· ,n−1 is the dual basis of (c )t=0,··· ,n−1 we obtain t j i j i j λtc = λtc , thus λt = 0. Hence, z = λic + λjc . Now using ∆(z) = z ⊗ c + c ⊗ z we obtain i j that λi + λj = 0, and thus z = λc − λc , for some λ ∈ k.

n For a positive integer n let Un(k) = {ω ∈ k | ω = 1} be the cyclic group of n-th roots of unity in k. The group Un(k) parameterizes the set of all matched pairs (H4, k[Cn], /, .).

Proposition 3.5.3 Let k be a ﬁeld of characteristic 6= 2, n a positive integer and Cn the cyclic group of order n. Then there exists a bijective correspondence between the set of all matched pairs (H4, k[Cn], ., /) and Un(k) such that the matched pair (., /) corresponding to an n-th root of unity ω ∈ Un(k) is given by:

/ 1 g x gx . 1 g x gx 1 1 1 0 0 1 1 g x gx c c c 0 0 c 1 g ωx ωgx ...... ck ck ck 0 0 ck 1 g ωkx ωkgx ...... cn−1 cn−1 cn−1 0 0 cn−1 1 g ωn−1x ωn−1gx

Proof: Let (H4, k[Cn], ., /) be a matched pair. It follows from the normalizing conditions (2.29) that ci . 1 = 1, 1 . z = z, 1 / z = ε(z)1, ci / 1 = ci for all i = 0, ··· , n − 1 and z ∈ H4. We prove now that c . g = g. Indeed, it follows from Lemma 3.5.1 that c . g is a group-like element in H4. Thus, using Lemma 3.5.2 we obtain that c . g ∈ {1, g}. If c . g = 1 using the fact that . is a left action we obtain ci . g = 1, for any positive integer i. In particular, g = 1 . g = cn . g = 1, contradiction. Thus, c . g = g and hence ci . g = g, for all i = 1, ··· , n − 1.

Now, x is an (1, g)-primitive element of H4. As c.g = g we obtain using Lemma 3.5.1 that c.x is an (1, g)-primitive element of H4. It follows from Lemma 3.5.2 that c . x = β − βg + ω x, for

164 3.5. EXAMPLES some β, ω ∈ k. We will ﬁrst prove that ωn = 1 and later on that β = 0. Indeed, by induction, having in mind that . is a left action we obtain:

i−1 i−1 X X ci . x = β ωj − β( ωj) g + ωi x j=0 j=0 for any positive integer i. In particular,

n−1 n−1 X X x = 1 . x = cn . x = β ωj − β( ωj)g + ωnx j=0 j=0

n Pn−1 j thus ω = 1 and β j=0 ω = 0. Similarly, we obtain that for any i = 1, ··· , n − 1 we have:

i−1 i−1 X X ci . (gx) = γ ζj − γ( ζj) g + ζi gx j=0 j=0

n Pn−1 j for some γ, ζ ∈ k, such that ζ = 1 and γ j=0 ζ = 0. At the end of the proof we will see that γ = β = 0 and ζ = ω. Meanwhile, using what we already know about the left action ., we shall prove that the other action / of the matched pair (H4, k[Cn], ., /) is necessarily the trivial action of H4 on k[Cn]. i First of all, using Lemma 3.5.1, we obtain that c / g is a group-like element in k[Cn], hence ci / g ∈ {1, c, ··· , cn−1}. We claim now that ci / g = ci, for all i = 1, ··· , n − 1. Indeed, suppose that there exist i 6= j such that ci / g = cj. Then, the compatibility condition (2.32) from the deﬁnition of a matched pair applied for (ci, x) takes the form:

ci / x ⊗ ci . 1 + ci / g ⊗ ci . x = ci / 1 ⊗ ci . x + ci / x ⊗ ci . g which is equivalent to:

i−1 i−1 X X ci / x ⊗ 1 + β ( ωt) cj ⊗ 1 − β ( ωt) cj ⊗ g + ωi cj ⊗ x = t=0 t=0 i−1 i−1 X X β ( ωt) ci ⊗ 1 − β ( ωt) ci ⊗ g + ωici ⊗ x + ci / x ⊗ g t=0 t=0

Now observe that the coefficient of cj ⊗ x in the left hand side of the equality is ωi while the coefficient of the same element in the right hand side of the equality is 0 if i 6= j. Since ωn = 1 we reached a contradiction. Thus ci / g = ci, for all i = 1, ··· , n − 1. Now, ci is a group like element of k[Cn] and x ∈ P1,g(H4). Thus, using Lemma 3.5.1 we obtain that i i c / x ∈ Pci/1, ci/g(k[Cn]) = Pci, ci (k[Cn]). Hence, it follows from Lemma 3.5.2 that c / x = 0. A similar argument shows that ci / (gx) = 0, for all i = 1, ··· , n − 1. Thus we have proved that / is the trivial action. In this case the compatibility conditions from the definition of a matched pair become simpler. More precisely the compatibility (2.30) becomes

165 CHAPTER 3. CLASSIFYING BICROSSED PRODUCTS OF QUANTUM GROUPS

h . (ab) = h(1) . a h(2) . b , for all h ∈ k[Cn], a and b ∈ H4, the compatibility (2.31) is trivially fulfilled while the compatibility (2.32) becomes h(1) ⊗ h(2) . a = h(2) ⊗ h(1) . a, for all h ∈ k[Cn], a ∈ H4 and is also automatically satisfied since k[C4] is cocommutative. Now, using the compatibility condition (2.30) in its simplified form c . (gx) = (c . g)(c . x), we obtain: γ − γg + ζ gx = g(β − βg + ωx) = −β + βg + ω gx and therefore, γ = −β and ζ = ω. Moreover, since 0 = c . x2 = (c . x)(c . x), we find that

0 = (β − βg + ωx)2 = 2β2 − 2β2g + 2βωx

As char(k) 6= 2 we obtain that β = 0 and, as a consequence, that ci . x = ωix and ci . (gx) = ωi(gx), for all i = 1, ··· , n − 1 and hence the left action . is also fully described. We are left to verify the rest of the compatibilities, and, since this is a routinely check, we omit it.

∼ We prove now that a Hopf algebra E factorizes through H4 and k[Cn] if and only if E = H4n, ω, where H4n, ω is a quantum group associated to any ω an n-th root of unity as described below.

Corollary 3.5.4 Let k be a ﬁeld of characteristic 6= 2 and n a positive integer. Then a Hopf ∼ algebra E factorizes through H4 and k[Cn] if and only if E = H4n, ω, for some ω ∈ Un(k), where we denote by H4n, ω the Hopf algebra generated by g, x and c subject to the relations:

g2 = cn = 1, x2 = 0, xg = −gx, cg = gc, cx = ω xc with the coalgebra structure and antipode given by:

∆(g) = g ⊗ g, ∆(c) = c ⊗ c, ∆(x) = x ⊗ 1 + g ⊗ x

ε(c) = ε(g) = 1, ε(x) = 0,S(c) = cn−1,S(g) = g, S(x) = −gx

H4n, ω is a pointed non-semisimple 4n-dimensional Hopf algebra and the canonical extension H4 ⊆ H4n, ω is rigid.

Proof: It follows from Proposition 3.5.3 and Theorem 3.1.3. The Hopf algebra H4n, ω is the bicrossed product H4 ./ k[Cn] associated to the matched pair given in Proposition 3.5.3. Indeed, up to canonical identiﬁcation, H4 ./ k[Cn] is generated as an algebra by g = g ./ 1, x = x ./ 1 and c = 1 ./ c. Hence,

c x = (1 ./ c)(x ./ 1) = c . x(1) ./ c / x(2) = ω x ./ c = ω (x ./ 1)(1 ./ c) = ω xc

H4 ./ k[Cn] is pointed by being generated by two group-likes and a primitive element [129, Lemma 1] and non-semisimple as H4 is non-semisimple. Now, k[Cn] is an H4-form of H4n, ω. As the right action / in Proposition 3.5.3 is the trivial action we obtain that any r-deformation of k[Cn] in the sense of Theorem 3.4.7 coincide with k[Cn]. Thus, the extension H4 ⊆ H4n, ω is rigid.

166 3.5. EXAMPLES

i i i i Remark 3.5.5 A k-basis in H4n, ω is given by {c , gc , xc , gxc | i = 0, ··· , n − 1}. We note that H4n, ω is the bicrossed product H4 ./ k[Cn] associated to the matched pair in which the right action / : k[Cn] ⊗ H4 → k[Cn] is the trivial action. Thus, H4n, ω is in fact the semi-direct product H4#k[Cn] associated to the left action . : k[Cn]⊗H4 → H4 given in Proposition 3.5.3. In particular, H4 is a normal Hopf subalgebra of H4n, ω. Moreover, the factorization index f [H4n, ω : H4] = 1, i.e. a Krull-Schmidt-Azumaya type result holds for the extension H4 ⊆ ∼ ∼ H4n, ω. That is, if we have two bicrossed product decompositions H4n, ω = H4 ./ k[Cn] = ∼ H4 ./ L, for some Hopf algebra L, then L = k[Cn].

In order to classify these Hopf algebras H4n, ω we still need one more elementary lemma:

Lemma 3.5.6 Let k be a ﬁeld of characteristic 6= 2, H4 the Sweedler’s 4-dimensional Hopf algebra and k[Cn] the group algebra of Cn. Then:

(1) u : H4 → H4 is a unitary coalgebra map if and only if u is the trivial morphism u(h) = ε(h)1, for all h ∈ H4, or there exists α, β, γ, δ ∈ k such that u(1) = 1, u(g) = g, u(x) = 1 α − α g + β x, and u(gx) = γ − γ g + δ gx. In particular, Hl,c(H4,H4) is the trivial group with only one element.

(2) u : H4 → H4 is a Hopf algebra morphism if and only if u is the trivial morphism u(h) = ε(h)1, for all h ∈ H4, or there exists β ∈ k such that u(1) = 1, u(g) = g, u(x) = β x, and ∼ ∗ u(gx) = β gx. In particular, Aut Hopf (H4) = k . (3) Assume that n is odd. Then:

(3a) p : H4 → k[Cn] is a morphism of Hopf algebras if and only if p is the trivial morphism: p(h) = ε(h)1, for all h ∈ H4.

(3b) r : k[Cn] → H4 is a morphism of Hopf algebras if and only if r is the trivial morphism: r(ci) = 1, for all i = 0, ··· , n − 1.

(4) Assume that n = 2m is even. Then:

(4a) p : H4 → k[Cn] is a morphism of Hopf algebras if and only if p is the trivial morphism or p is given by: p(1) = 1, p(g) = cm, p(x) = p(gx) = 0.

(4b) r : k[Cn] → H4 is a morphism of Hopf algebras if and only if r is the trivial morphism or r is given by r(ci) = gi, for all i = 0, ··· , n − 1.

Proof: (1) Let u : H4 → H4 be a unitary coalgebra map. Then u(g) ∈ G(H4) = {1, g} and u(x) ∈ P1, u(g)(H4) respectively u(gx) ∈ Pu(g), 1(H4). By applying Lemma 3.5.2 we arrive at the desired conclusion: if u(g) = 1, then u is the trivial morphism, while if u(g) = g, then u has the second form. A little computation shows that among all these morphisms only the trivial one satisﬁes the cocycle condition (2.62).

(2) It follows from (1). If u : H4 → H4 is the non-trivial morphism, then considering the relations on the generators of H4 we obtain that u is also an algebra map if and only if α = γ = 0 and δ = β. The ﬁnal assertion is obvious.

167 CHAPTER 3. CLASSIFYING BICROSSED PRODUCTS OF QUANTUM GROUPS

(3) and (4) Let now p : H4 → k[Cn] be a Hopf algebra map. Then, p(g) is a group-like element i in k[Cn], i.e. p(g) = c , for some i ∈ {0, 1, ..., n − 1}. It follows from 1 = p(g2) = p(g)2 = c2i that n | 2i. If n is odd, then n | i, hence i = 0 i.e. p(g) = 1. If n = 2m is even, then m | i, hence i = 0 or i = m. Thus, p(g) = 1 or p(g) = cm. On the other hand, x is an (1, g)-primitive element of H4, hence p(x) ∈ P1, p(g)(k[Cn]). Using Lemma 3.5.2 we obtain that p(x) = 0 if p(g) = 1, and p(x) = λ − λcm if p(g) = cm. In the last case we have: 0 = p(x2) = p(x)2 = (λ − λcm)2 = 2λ2 − 2λ2cm therefore λ = 0 i.e. p(x) = 0. A similar discussion describes Hopf algebra maps r : k[Cn] → H4 and we are done.

We shall give now the necessary and sufficient conditions for two Hopf algebras H4n, ω and H4n, ω0 to be isomorphic. The classification Theorem 3.5.7 below depends on the structure of Un(k). We denote by ν(n) = |Un(k)|, the order of the cyclic group Un(k) and we shall fix ξ a l 0 t generator of Un(k). Let ω = ξ , ω = ξ be two arbitrary n-th roots of unity, for some positive integers l, t ∈ { 0, ··· , ν(n) − 1 }.

Theorem 3.5.7 Let k be a ﬁeld of characteristic 6= 2, n a positive integer, ξ a generator of the group Un(k) of order ν(n) and l, t ∈ { 0, ··· , ν(n) − 1 }. Then:

(1) Assume that one of the positive integers n or ν(n) is odd. Then the Hopf algebras H4n, ξl and H4n, ξt are isomorphic if and only if there exists s ∈ {0, 1, ··· , n − 1} such that gcd (s, n) = 1 and ν(n) | l − ts.

(2) Assume that n and ν(n) are both even. Then the Hopf algebras H4n, ξl and H4n, ξt are isomorphic if and only if there exists s ∈ {0, 1, ··· , n − 1} such that gcd (s, n) = 1 and one of the following conditions hold: ν(n) | l − ts or 2(l − ts) = ν(n) q, for some odd integer q.

Proof: We shall prove more. In fact we will describe the set of all possible Hopf algebra iso- l morphisms between H4n, ξl and H4n, ξt . We denote by H4n, ξl := H4# k[Cn] (resp. H4n, ξt := t H4# k[Cn]) the semidirect product implemented by the left action . : k[Cn] ⊗ H4 → H4, c . x = ξlx (resp. c .0 x = ξtx) from Proposition 3.5.3. We recall from Corollary 3.2.3 that ψ : H4n, ξl → H4n, ξt is an isomorphism of Hopf algebras if and only if there exists two unitary coalgebra maps u : H4 → H4, r : k[Cn] → H4 and two morphisms of Hopf algebras p : H4 → k[Cn], v : k[Cn] → k[Cn] such that for any a, b ∈ H4, g, h ∈ k[Cn] we have

u(a(1)) ⊗ p(a(2)) = u(a(2)) ⊗ p(a(1)) (3.74)

r(h(1)) ⊗ v(h(2)) = r(h(2)) ⊗ v(h(1)) (3.75) 0 u(ab) = u(a(1)) p(a(2)) . u(b) (3.76) 0 r(hg) = r(h(1)) v(h(2)) . r(g) (3.77) 0 0 r(h(1)) v(h(2)) . u(b) = u(h(1) . b(1)) p(h(2) . b(2)) . r(h(3)) (3.78)

v(h) p(b) = p(h(1) . b) v(h(2)) (3.79)

168 3.5. EXAMPLES

and ψ = ψ(u,r,p,v) : H4n, ξl → H4n, ξt is given by:

0 ψ(a#h) = u(a(1)) p(a(2)) . r(h(1)) # p(a(3)) v(h(2)) (3.80) for all a ∈ H4 and h ∈ k[Cn]. In what follows we describe completely all quadruples (u, r, p, v) which satisfy the compatibility conditions (3.74)-(3.79) such that ψ = ψ(u,r,p,v) given by (3.80) is bijective. First, we should notice that (3.75) holds trivially as k[Cn] is cocommutative. Also any morphism of Hopf algebras v : k[Cn] → k[Cn] is given by

s v : k[Cn] → k[Cn], v(c) = c (3.81) for some s = 0, ··· , n − 1. For future use, we note that such a morphism v is bijective if and only if (s, n) = 1.

Next we shall prove simultaneously that any Hopf algebra map p : H4 → k[Cn] of a such quadruple (u, r, p, v) is the trivial morphism

p : H4 → k[Cn], p(z) = ε(z)1 (3.82) for any z ∈ H4 and any unitary coalgebra morphism u : H4 → H4 of a such quadruple (u, r, p, v) is given by

u : H4 → H4, u(1) = 1, u(g) = g, u(x) = γ x, u(gx) = γ gx (3.83) for some non-zero scalar γ ∈ k∗. Indeed, it follows from (3a) and (4a) of Lemma 3.5.6 that p(x) = 0 and p(gx) = 0. Now, using the normalizing conditions (2.29), the fact that r, v are unitary maps and p is a coalgebra map we obtain from (3.80) that

ψ(x#1) = u(x(1))#p(x(2)) = u(x)#1 + u(g)#p(x) = u(x)#1

As ψ has to be an isomorphism and x is an element of the basis, we obtain that u(x) 6= 0. It follows from (1) of Lemma 3.5.6 that u(x) = α − α g + γ x, for some α, γ ∈ k. Now, by applying (3.74) for a = x, we obtain:

(α − α g + γ x) ⊗ 1 = (α − α g + γ x) ⊗ p(g)

As u(x) 6= 0, we must have p(g) = 1 and therefore p : H4 → k[Cn] is the trivial morphism. Condition (3.76) is then equivalent to u : H4 → H4 being an algebra map, i.e. u is a morphism of Hopf algebras. Using (2) of Lemma 3.5.6 we obtain that u is given by (3.83), where γ ∈ k∗, since u(x) 6= 0. Thus, we fully described the maps u, v and p in the quadruple (u, r, p, v). Furthermore, since p is the trivial morphism the compatibility conditions (3.74) and (3.79) are trivially fulﬁlled.

We focus now on the unitary morphism of coalgebras r : k[Cn] → H4 in a such quadruple. As i r is a coalgebra map we have that r(c ) ∈ G(H4) = {1, g}, for all i = 0, ··· , n − 1. Since ci .0 z = z, for all i = 0, ··· , n − 1 and z ∈ {1, g}, it follows that (3.77) is equivalent to r : k[Cn] → H4 being an algebra map, that is r : k[Cn] → H4 is a Hopf algebra map. In

169 CHAPTER 3. CLASSIFYING BICROSSED PRODUCTS OF QUANTUM GROUPS conclusion, taking into account that p is the trivial morphism, we proved so far that any potential isomorphism ψ : H4n, ξl → H4n, ξt is deﬁned by the formula l t ψ : H4 # k[Cn] → H4 # k[Cn], ψ(a#h) = u(a) r(h(1))# v(h(2)) (3.84) for all a ∈ H4 and h ∈ k[Cn], where u : H4 → H4 is the isomorphism of Hopf algebras given by (3.83), v : k[Cn] → k[Cn] is the Hopf algebra map given by (3.81) and r : k[Cn] → H4 is a morphism of Hopf algebras satisfying the compatibility condition (3.78) in its simpliﬁed form, namely 0 r(h(1)) v(h(2)) . u(b) = u(h(1) . b) r(h(2)) (3.85) for all h ∈ k[Cn] and b ∈ H4.

Finally, it remains to describe when there exists a Hopf algebra map r : k[Cn] → H4 satisfying the compatibility condition (3.85) such that ψ given by (3.84) is bijective. According to Lemma 3.5.6 we distinguish two cases. Suppose first that n is odd. Then using (3b) of Lemma 3.5.6 we obtain that r : k[Cn] → H4 is also the trivial morphism namely, r(ci) = ε(ci)1 = 1, for all i ∈ 0, 1, ··· , n − 1. Hence, ψ given by (3.84) takes the simplified form ψ(a#h) = u(a)#v(h) and ψ is an isomorphism if and only if v is bijective, i.e. if and only if (s, n) = 1. Moreover, the compatibility condition (3.85) becomes: v(h) .0 u(a) = u(h . a) (3.86) for all a ∈ H4 and h ∈ k[Cn]. Applying the above compatibility for a = x and h = c we obtain ξl−st = 1, thus ν(n) | (l − st). Moreover, it is easy to see that if ν(n) | (l − st), then (3.86) holds for any a ∈ H4 and h ∈ k[Cn]. Finally, suppose now that n is even and use (4b) of Lemma 3.5.6. If r is the trivial morphism then the proof follows exactly as in the above odd case. Assume now that r(c) = g. Hence, ψ is n−1 given by ψ(a#h) = u(a)r(h)#v(h), for all a ∈ H4 and h ∈ {1, c, ··· , c }. It is easy to see that ψ is an isomorphism if and only if v is an isomorphism. i.e. if and only if (s, n) = 1. Now, we observe that the compatibility condition (3.85) is equivalent to ξl−ts = −1. Indeed, (3.85) applied for a = x and h = c gives precisely ξl−ts = −1. Conversely, if (3.85) holds for a = x and h = c, then it is straightforward to see that it is fulfilled for any a ∈ H4 and any h ∈ k[Cn]. We shall prove now that ξl−ts = −1 if and only if ν(n) is even and 2(l − ts) = ν(n) q, for some odd integer q and this finishes the proof. Indeed, since Char(k) 6= 2 and ξν(n) = 1, the equation ξl−ts = −1 is possible only if ν(n) is even. Consider ν(n) = 2m, for a positive integer m. As ξ2m = 1, we obtain ξm = −1, otherwise using that 2m is the order of ξ we will obtain 2m|m, contradiction. Assume first that ξl−ts = −1. Then, ξ2(l−ts) = 1 and hence ν(n) | 2(l − ts), that is l − ts = mq, for some integer l−ts m q q q. Thus, −1 = ξ = (ξ ) = (−1) , hence q is odd. Conversely, is straightforward.

Corollary 3.5.8 Let k be a ﬁeld of characteristic 6= 2, n a positive integer and ξ a generator of the group Un(k). Then, H4n, ξl is isomorphic to H4n, ξ, for any positive integer l ∈ {1, ··· , ν(n) − 1} such that gcd (l, n) = 1.

In particular, if n = p is a prime odd number, then a Hopf algebra factorizes through H4 and k[Cp] if and only if it is isomorphic to H4n,1 or H4n,ξ, where ξ is a generator of Up(k).

170 3.5. EXAMPLES

Proof: We apply the Theorem 3.5.7 for t = 1 by taking s := l in its statement.

In what follows we continue our investigation in order to indicate precisely the number of all types of isomorphisms of Hopf algebras which factorize through H4 and k[Cn]. For this we need the following lemma in the proof of which we use Dirichlet’s theorem on primes in an arithmetical progression.

Lemma 3.5.9 Let n and m be two positive integers such that m | n and ϕ : Zn → Zm the canonical projection ϕ(a + nZ) = a + mZ, for all a ∈ Z. Then ϕ (U(Zn)) = U(Zm).

Proof: Consider a + mZ ∈ U(Zm). Thus, gcd(a, m) = 1. The Dirichlet’s theorem [51, Theorem 8] ensure the fact that in the set {a + km|k ∈ Z} there exists inﬁnitely many primes. In particular, there exists a prime number p ∈ {a + km|k ∈ Z} such that p - n. As ϕ(p + nZ) = a + mZ, and p + nZ ∈ U(Zn), we deduce that U(Zm) ⊆ ϕ (U(Zn)). Obviously, ϕ (U(Zn)) ⊆ U(Zm), hence our claim.

We shall describe and count the set of types of Hopf algebras that factorize through H4 and k[Cn] .

Theorem 3.5.10 Let k be a ﬁeld of characteristic 6= 2, n a positive integer, ξ a generator of α1 αr Un(k) and let ν(n) = p1 ··· pr be the prime decomposition of ν(n). Then:

(1) There exists an isomorphism of Hopf algebras H4n, ξt ' H4n, ξgcd(t, ν(n)) , for all t = 0, ··· , ν(n) − 1. In particular, H4n, ξi ' H4n, ξn−i , for any i = 0, ··· , ν(n) − 1.

(2) Assume that ν(n) is odd. Then the set of types of Hopf algebras that factorize through H4 and k[Cn] is in bijection with the set of all Hopf algebras H4n, ξd , where d running over all positive divisors of ν(n). In particular, the number of types of such Hopf algebras is

(α1 + 1)(α2 + 1) ··· (αr + 1)

α1 α2 αr (3) Assume that ν(n) = 2 p2 ··· pr is even. Then the set of types of Hopf algebras that factorize through H4 and k[Cn] is in bijection with the set of all Hopf algebras H4n, ξd , where ν(n) d is running over all positive divisors of . In particular, the number of types of such Hopf 2 algebras is α1(α2 + 1) ··· (αr + 1)

Proof: (1) Let d = gcd(t, ν(n)). Consider two positive integers a and m such that t = da and ν(n) = dm. Since gcd(a, m) = 1 and m | ν(n) | n, we obtain from Lemma 3.5.9 that there exists s ∈ {0, 1, ..., n − 1} such that gcd(s, n) = 1 and m | a − s. Multiplying the last relation by d, we obtain ν(n) | t − ds. This, together with Theorem 3.5.7, proves that H4n, ξt ' H4n, ξd . Since ν(n)|n, the last statement follows from the ﬁrst part using that gcd(i, ν(n)) = gcd(n − i, ν(n)), for any i = 1, ··· , ν(n) − 1.

171 CHAPTER 3. CLASSIFYING BICROSSED PRODUCTS OF QUANTUM GROUPS

(2) In the ﬁrst part we have proved that any Hopf algebra H4n, ξt is isomorphic to H4n, ξd , for some divisor d of ν(n). We will show now that if d1 and d2 are two distinct positive divisors of

ν(n) then H4n,ξd1 and H4n,ξd2 are not isomorphic. This will prove our claim.

Let therefore d1 and d2 be two distinct positive divisors of ν(n). Suppose that H4n,ξd1 ' H4n,ξd2 . By Theorem 3.5.7 there exists then s ∈ {0, 1, ..., n − 1} such that gcd(s, n) = 1 and ν(n) | d1 − sd2. Since d1 and d2 are distinct, they differ by the exponent of a prime number, say p. Let α and β be the respective exponents of d1 and d2. We do not restrict the generality by assuming that α > β. Since ν(n) d d | 1 − s 2 pβ pβ pβ ν(n) d d p | , p | 1 , and p s 2 , we have arrived at the desired contradiction. pβ pβ - pβ ν(n) (3) We ﬁrst prove that if d and d are two distinct positive divisors of then H d 1 2 2 4n,ξ 1 H4n,ξd2 . Then we prove that if d1 is a positive divisor of ν(n) then there exists a positive divisor ν(n) of , d , such that H d =∼ H d . This, together with assertion (1) will ﬁnish the proof. 2 2 4n,ξ 1 4n,ξ 2 ν(n) Suppose d and d are two distinct positive divisors of . Then, if s ∈ {0, 1, ..., n − 1} and 1 2 2 ν(n) gcd(s, n) = 1, d − d s. Indeed, d and d being distinct they differ by the exponent 2 - 1 2 1 2 of a prime number, p. Let α and β be the respective exponents of d1 and d2. We may assume, ν(n) d d without loss of generality that α > β. Since p | , p | 1 , and p s 2 (recall that ν(n) | n, 2pβ pβ - pβ and gcd(s, n) = 1) we cannot have ν(n) d d | 1 − s 2 2pβ pβ pβ ν(n) hence, neither | d −d s. This being the case, we deduce from Theorem 3.5.7 that H d 2 1 2 4n,ξ 1 and H4n,ξd2 are not isomorphic.

For the second claim, consider d1 a positive divisor of ν(n). If d1 = ν(n), then H4n,ξd1 = ∼ H4n,ξ0 = H ν(n) , as it results from Theorem 3.5.7 observing that: 4n,ξ 2 ν(n) 2 − 1 · 0 = ν(n) · 1 2 ν(n) ν(n) ν(n) Thus d = in this case. If d | , we take d = d . If d 6= ν(n) and d , we 2 2 1 2 2 1 1 1 - 2 d1 ν(n) α α take d2 = . Indeed, 2 - , hence ν(n) = 2 u and d1 = 2 v, for some positive integer α 2 d1 and odd integers u and v such that v | u. Let −q be the product of all prime divisors of n that do u u u n not divide 2 , and s = 2 − q. Then q is an odd integer, − q ≤ , and gcd(s, n) = 1. Also, v v v 2 u n s = 2 − q ≤ 2 + ≤ n − 1 v 2

172 3.5. EXAMPLES

u d1 as soon as n ≥ 6. Multiplying s = 2 − q by d2 = we ﬁnd that 2(d1 − sd2) = ν(n)q. ∼v 2 Therefore, when n ≥ 6, we have H4n,ξd1 = H4n,ξd2 , by virtue of Theorem 3.5.7 . If n < 6 then ν(n) = 2 or ν(n) = 4, cases in which there is nothing more to prove.

Example 3.5.11 A straightforward computation based on Theorem 3.5.10 shows the following: if ν(n) = 2 then there is only one type of isomorphism, namely the tensor product while if ν(n) ∈ {3, 4, 5, 6, 7} there are two types of isomorphisms of bicrossed products between H4 and k[Cn] as follwos: H4n, 1 and H4n, ξ. If ν(n) = 8 or ν(n) = 9 then there are three types of isomorphisms of bicrossed products between H4 and k[Cn] namely H4n, 1, H4n, ξ and H4n, ξ2 if ν(n) = 8 and respectively H4n, 1, H4n, ξ and H4n, ξ3 for ν(n) = 9.

In what follows we will iterate this construction by considering a family of matched pairs l (k[Cn],H4n, ξt ,/ ,.) such that the Hopf algebra H4n, ξt−lp will appear as r-deformations in the sense of Theorem 3.4.7 of H4n, ξt . The purpose is that of constructing Hopf algebra extensions E/A with a given factorization index.

Theorem 3.5.12 Let k be a ﬁeld of characteristic 6= 2, n a positive integer, ξ a generator of n Un(k), t ∈ {0, 1, ..., ν(n) − 1} and Cn = hd | d = 1i the cyclic group of order n. Then: l (1) For any l ∈ {0, 1, ··· , ν(n) − 1} there exists a matched pair (k[Cn],H4n, ξt , ., / ), where l . : H4n, ξt ⊗ k[Cn] → k[Cn] is the trivial action and the right action / : H4n, ξt ⊗ k[Cn] → H4n, ξt is given by:

ci /l dk = ci, (gci) /l dk = gci, (xci) /l dk = ξlk xci, (gxci) /l dk = ξlk gxci (3.87) for all i, k = 0, 1, ··· , n − 1. (2) The (., /l)-cocycles are algebra maps deﬁned as follows:

p rp : H4n, ξt → k[Cn], rp(g) = 1, rp(c) = d , rp(x) = 0

where p ∈ {0, 1, ··· , n − 1} and the rp-deformation of H4n, ξt is H4n, ξt−lp , i.e. (H4n, ξt )rp = H4n, ξt−lp .

Proof: (1) First, notice that since . is the trivial action and k[Cn] is cocomutative then the compatibility condition (2.32) is trivially fulﬁlled. Moreover, (2.31) becomes:

l l l (yz) / a = (y / a(1))(z / a(2)) (3.88) for all y, z ∈ H4n, ξt and a ∈ k[Cn]. Since we have:

ci /l dk = ci, g /l dk = g, x /l dk = ξlkx then it is straightforward to see that (3.88) indeed holds.

173 CHAPTER 3. CLASSIFYING BICROSSED PRODUCTS OF QUANTUM GROUPS

l Now we prove that / : H4n, ξt ⊗ k[Cn] → H4n, ξt is a coalgebra map. Indeed, it is straightforward to see that ∆(ci /l dk) = ci /l dk ⊗ ci /l dk and ∆(gci /l dk) = gci /l dk ⊗ gci /l dk for all i, k = 0, 1, ··· , n − 1. Furthermore, we have:

i l k i l k i l k ∆(xc / d ) = (xc / d )(1) ⊗ (xc / d )(2) lk i i = ξ (xc )(1) ⊗ (xc )(2) = ξlkxci ⊗ ci + ξlkgci ⊗ xci = xci /l dk ⊗ ci /l dk + gci /l dk ⊗ xci /l dk i l k i l k = (xc )(1) / d ⊗ (xc )(2) / d

i l k i l k i l k ∆(gxc / d ) = (gxc / d )(1) ⊗ (gxc / d )(2) lk i i = ξ (gxc )(1) ⊗ (gxc )(2) = ξlkgxci ⊗ gci + ξlkci ⊗ gxci = gxci /l dk ⊗ gci /l dk + ci /l dk ⊗ gxci /l dk i l k i l k = (gxc )(1) / d ⊗ (gxc )(2) / d

l Finally, we only need to prove that the action / respects the relations in k[Cn], respectively H4n, ξt . For instance, we have:

xci /l dn = (xci /l dn−1) /l d = ξl(n−1)xci /l d = ξlnxci = xci

xg /l dk = (x /l dk)(g /l dk) = ξlkxg = −ξlkgx = −gx /l dk cx /l dk = (c /l dk)(x /l dk) = ξlkcx = ξlkξtxc = ξtxc /l dk Proving that the rest of the compatibilities also hold is a routinely check. l i i (2) Let r : H4n, ξt → k[Cn] be a (., / )-cocycle. By applying (2.62) for xc and gxc where i = 0, 1, ··· , n − 1 we get:

r(ci) ⊗ xci + r(xci) ⊗ gci = r(xci) ⊗ ci + r(gci) ⊗ xci

r(gci) ⊗ gxci + r(gxci) ⊗ ci = r(gxci) ⊗ gci + r(ci) ⊗ gxci Hence, it follows that r(xci) = r(gxci) = 0 and r(ci) = r(gci) for all i = 0, 1, ··· , n − 1. In particular we have r(g) = 1 and r(x) = 0. Moreover, since r is also a coalgebra map then r(c) p is a grouplike element from k[Cn]. Consider r(c) = d for some p = 0, 1, ··· , n − 1. For the rest of the proof we denote this map by rp. As . is the trivial action then the compatibility condition (3.63) simpliﬁes to:

l rp y / r(z(1)) z(2) = rp(y)rp(z) (3.89)

i j for all y, z ∈ H4n, ξt . By applying (3.89) for c and c , where i, j = 0, 1, ··· , n − 1 we get i+j i j rp(c ) = rp(c )rp(c ). Hence, we have

i i ip rp(c ) = rp(gc ) = d (3.90)

174 3.5. EXAMPLES

i i for all i = 0, 1, ··· , n − 1. Now using (3.90) and the fact that rp(xc ) = rp(gxc ) = 0 for all i = 0, 1, ··· , n − 1 we can easily prove that rp is an algebra map. For instance, we have: i j i j rp(c gxc ) = 0 = rp(c )rp(gxc ) i j i j rp(xc c ) = 0 = rp(xc )rp(c ) i j i j rp(xc xc ) = 0 = rp(xc )rp(xc ) i j i+j p(i+j) pi pj i j rp(c gc ) = rp(gc ) = d = d d = rp(c )rp(gc ) i j i+j p(i+j) pi pj i j rp(gc gc ) = rp(c ) = d = d d = rp(gc )rp(gc ) for all i, j = 0, 1, ··· , n − 1. It is straightforward to see that the rest of the compatibilities also hold and rp is indeed an algebra map. Finally, we are left to prove that (3.89) holds. For instance we have:

i l j j i l pj j i+j i j rp gc / rp(c ) c = rp gc / d c = rp(gc ) = rp(gc )rp(c )

i l j j i l pj j lpj i+j i j rp xc / rp(c ) c = rp xc / d c = rp(ξ xc ) = 0 = rp(xc )rp(c )

l i i l i i l i i i rp y/ rp((xc )(1)) (xc )(2) = rp y/ rp(xc ) c +rp y/ rp(gc ) xc = 0 = rp(y)rp(xc ) for all i, j = 0, 1, ··· , n − 1 and y ∈ H4n, ξt . It is straightforward to check that (3.89) holds for the remaining elements in the k-basis of H4n, ξt .

Finally, we recall that the algebra structure of (H4n, ξt )rp is given by (3.65). Thus, in (H4n, ξt )rp we have: l l 2 g • g = g / rp(g) g = (g / 1)g = g = 1 x • x = (x /l r(x)) + (x /l r(g))x = x2 = 0 n−1 n−1 l n−1 l p n−1 n c • c = c / rp(c) c = c / c c = c c = c = 1 g • x = (g /l r(x)) + (g /l r(g))x = gx = −xg = −(x /l r(g))g = −x • g l l t t−lp lp t−lp l t−lp c • x = (c / r(x)) + (c / r(g))x = cx = ξ xc = ξ (ξ xc) = ξ (x / rp(c))c = ξ x • c

This shows that (H4n, ξt )rp = H4n, ξt−lp .

l The bicrossed product k[Cn] ./ H4n, ξt associated to the matched pair of Theorem 3.5.12 is the 4n2-dimensional Hopf algebra generated as an algebra by g, x, c and d subject to the relations g2 = cn = dn = 1, x2 = 0, cg = gc, cd = dc, gd = dg, xg = −gx, cx = ξt xc, xd = ξl dx with the coalgebra structure given such that g, c, d are group-like elements and x is an (1, g)- primitive element. We denote by H4n2, ξ, t, l these family of quantum groups, for any l, t ∈ {0, 1, ··· , ν(n) − 1} and ξ a generator of order ν(n) of the group Un(k). n We view H4n2, ξ, t, l as a Hopf algebra extension of the group algebra k[Cn] = khd | d = 1i and of course H4n, ξt is a factorization k[Cn]-form of H4n2, ξ, t, l. If we let l = 1, the next corollary computes the factorization index of this extension.

175 CHAPTER 3. CLASSIFYING BICROSSED PRODUCTS OF QUANTUM GROUPS

Corollary 3.5.13 Let k be a ﬁeld of characteristic 6= 2, n a positive integer, ξ a generator of 1 1 Un(k) and (k[Cn],H4n, ξν(n)−1 , ., / ) a matched pair, where . is the trivial action and / is given by (3.87) for l = 1. Then:

p 1) (H4n, ξν(n)−1 )rp = H4n, ξν(n)−1−p , for all p = 0, 1, ··· , ν(n) − 1. Thus, any H4n, ξ appears 1 as a deformation of H4n, ξν(n)−1 via some (., / )-cocycle rp. α1 αr 2) Assume that ν(n) is odd and ν(n) = p1 ··· pr is the prime decomposition of ν(n). Then we have (α1 + 1)(α2 + 1)...(αr + 1) non-isomorphic deformations of H4n, ξν(n)−1 and thus f [H4n2, ξ, t, 1 : k[Cn]] = (α1 + 1)(α2 + 1)...(αr + 1). α1 α2 αr 3) Assume that ν(n) is even and ν(n) = 2 p2 ··· pr is the prime decomposition of ν(n). Then we have α1(α2 + 1)...(αr + 1) non-isomorphic deformations of H4n, ξν(n)−1 and thus f [H4n2, ξ, t, 1 : k[Cn]] = α1(α2 + 1)...(αr + 1).

Proof: 1) It follows by applying 3) of Theorem 3.5.12 for l = 1 and t = ν(n) − 1. As 1 any H4n, ξp appears as a deformation of H4n, ξν(n)−1 via some (., / )-cocycle rp, the last two statements are just easy consequences of Theorem 3.5.10.

We conclude the discussion on these family of quantum groups by describing the group of Hopf algebra automorphisms of H4n, ξt . For any t = 0, 1, ··· , ν(n) − 1 we deﬁne:

Ut(Zn) := {sˆ ∈ U(Zn); ν(n) | t(s − 1)} ˆ Vt(Zn) := {l ∈ U(Zn) ; 2t(l − 1) = ν(n)q, for some odd integer q}

Uft(Zn) := Ut(Zn) ∪ Vt(Zn)

Corollary 3.5.14 Let k be a ﬁeld of characteristic 6= 2, n a positive integer, ξ a generator of the group Un(k) of order ν(n) and t = 0, 1, ··· , ν(n) − 1. Then Ut(Zn), Uft(Zn) are subgroups of the group of units U(Zn) and there exists an isomorphism of groups

∗ ∗ Aut Hopf (H4n, ξt ) ' k × Ut(Zn) or Aut Hopf (H4n, ξt ) ' k × Uft(Zn) the latter case holds if and only if n and ν(n) are both even.

Proof: To start with we prove that Ut(Zn) and Uft(Zn) are subgroups of U(Zn). Indeed, take s, p ∈ Ut(Zn). Then ν(n) | t(s − 1), ν(n) | t(p − 1). It follows that ν(n) | t(s − 1)(p − 1) which is equivalent to ν(n) | t(sp − p − s + 1). Moreover, we also have ν(n) | t(s + p − 2). Thus, we get ν(n) | t(sp − 1). Hence, we obtain sp ∈ Ut(Zn) as desired. Take now s ∈ Ut(Zn) and l ∈ Vt(Zn). Then 2t(l − 1) = ν(n)q, for an odd integer q and t(s − 1) = ν(n)w, for some integer w. It follows that t(s − 1)(l − 1) = ν(n)w(l − 1) which is equivalent to t(sl − s − l + 1) = ν(n)w(l − 1). Moreover, we also have 2t(s + l − 2) = ν(n)(2w + q). Hence we get 2t(sl − 1) = ν(n)2wl + q and since 2wl + q is odd it follows that sp ∈ Vt(Zn) ⊂ Uft(Zn) as desired. Finally, if l, r ∈ Vt(Zn) then 2t(l − 1) = ν(n)q1 and 2t(r − 1) = ν(n)q2, for some odd integers q1, q2. It is straightforward to see that 2t(lr − 1) =

176 3.5. EXAMPLES

ν(n) q1(r−1)+(q1 +q2) and since q1(r−1)+(q1 +q2) is an even integer we get ν(n) | (lr−1) and thus lr ∈ Ut(Zn) ⊂ Uft(Zn). Suppose ﬁrst that n is odd. Then according to the proof of Theorem 3.5.7, any Hopf algebra automorphism of H4n, ξt has the following form:

ψγ, s : H4n, ξt → H4n, ξt , ψγ, s(a#h) = u(a)#v(h) (3.91) where the Hopf algebra maps u : H4 → H4 and v : k[Cn] → k[Cn] are deﬁned as follows:

u(1) = 1, u(g) = g, u(x) = γ x, v(c) = cs

∗ for some non-zero scalar γ ∈ k and s ∈ Ut(Zn) . By a straightforward computation it can be seen that ψγ, s ◦ ψζ, s0 = ψγζ, ss0 . Then, the following map is an isomorphism of groups:

∗ Γ : Aut Hopf (H4n, ξt ) → k × Ut(Zn), Γ(ψγ, s) = (γ, s)

Assume now that n is even. Then, again by the proof of Theorem 3.5.7, we have two types of Hopf algebra automorphism for H4n, ξt . The ﬁrst such type of automorphisms is given by the maps ψγ, s deﬁned in (3.91) while the second one is given by:

ϕσ, l : H4n, ξt → H4n, ξt , ϕσ, l(a#h) = u(a)r(h(1))#v(h(2))

∗ for some σ ∈ k and l ∈ Vt(Zn), where the Hopf algebra maps u : H4 → H4, r : k[Cn] → H4 and v : k[Cn] → k[Cn] are deﬁned as follows:

u(1) = 1, u(g) = g, u(x) = σ x, r(c) = g, v(c) = cl

∗ Now deﬁne Γe : Aut Hopf (H4n, ξt ) → k × Uft(Zn) as follows:

∗ ∗ Γ(e ψγ, s) = (γ, s) ∈ k × Ut(Zn), Γ(e ϕσ, l) = (σ, l) ∈ k × Vt(Zn)

∗ for all γ, σ ∈ k and s ∈ Ut(Zn), l ∈ Vt(Zn). As Ut(Zn) ∩ Vt(Zn) = ∅ it can be easily seen that Γe is well deﬁned and bijective. Moreover, Γe is also a morphism of groups:

0 0 Γ(e ψγ, s ◦ ψζ, s0 ) = Γ(e ψγζ, ss0 ) = (γζ, ss ) = (γ, s)(ζ, s ) = Γ(e ψγ, s)Γ(e ψζ, s0 )

Γ(e ψγ, s ◦ ϕσ, l) = Γ(e ϕγσ, sl) = (γσ, sl) = (γ, s)(σ, l) = Γ(e ψγ, s)Γ(e ϕσ, l) 0 0 Γ(e ϕσ, l ◦ ϕτ, l0 ) = Γ(e ψστ, ll0 ) = (στ, ll ) = (σ, l)(τ, l ) = Γ(e ϕσ, l)Γ(e ϕτ, l0 )

2 2 Let now, K = C2 × C2 = ha, b | a = b = 1, ab = bai be the Klein’s group. Analogous with Corollary 3.5.4 we can prove the following classiﬁcation result. It shows that we can have various matched pairs for two given Hopf algebras, but all associated bicrossed products are isomorphic to the trivial one, namely the tensor product of Hopf algebras.

177 CHAPTER 3. CLASSIFYING BICROSSED PRODUCTS OF QUANTUM GROUPS

Proposition 3.5.15 Let k be a ﬁeld of characteristic 6= 2. Then a Hopf algebra E factorizes through H4 and k[C2 × C2] if and only if E is isomorphic to the tensor product of Hopf algebras H4 ⊗ k[C2 × C2].

Proof: Let (H4, k[C2 × C2], ., /) be a matched pair. We start by making some simple observations which will simplify our computations. First, we should notice that for all group-like elements h ∈ k[C2 × C2] we have h . g 6= 1 for otherwise we would obtain:

g = h2 . g = h . (h . g) = h . 1 = 1 which is a contradiction. In the same manner it can be proved that for all group-like elements h ∈ k[C2 × C2] different from 1 we have h / g 6= 1. Moreover, if h1, h2 are two distinct group-like elements of k[C2 × C2] then h1 / g 6= h2 / g. Indeed, if h1 / g = h2 / g we obtain:

2 2 h1 = h1 / g = (h1 / g) / g = (h2 / g) / g = h2 / g = h2 which is again a contradiction. Also, by a straightforward computation it can be proved that for all group-like elements h1, h2 of k[C2×C2] we have Ph1, h2 (k[C2×C2]) = {λ(h1−h2) | λ ∈ k}. Now using the above remarks and Lemma 3.5.1 we obtain:

a . g = b . g = g, a / g, b / g ∈ {a, b, ab}

a / x ∈ {λ(a − a / g) | λ ∈ k}, b / x ∈ {σ(b − b / g) | σ ∈ k} a . x, b . x ∈ {α − αg + βx | α, β ∈ k} Consider a . x = α − αg + βx for some α, β ∈ k. By imposing the condition x = a2 . x we obtain: x = a2 . x = a . (a . x) = a . (α − αg + βx) = α(1 + β) − α(1 + β)g + β2x. Thus a . x is either equal to x or to α − αg − x for some α ∈ k. Of course, this also holds for b . x. Suppose ﬁrst that a / g = ab. Then we have:

(2.31) a = ab / g = a / (b . g)(b / g) = ab(b / g)

Thus, we obtained b / g = b. Moreover, we also have a / x = λ(a − ab) and b / x = 0 where λ ∈ k. Now if we apply (2.32) for a ∈ [C2 × C2] and x ∈ H4 we get:

λa ⊗ 1 − λab ⊗ 1 + ab ⊗ a . x = a ⊗ a . x + λa ⊗ g − λab ⊗ g

Observe that if a . x = x the above relation becomes:

λa ⊗ 1 − λab ⊗ 1 + ab ⊗ x = a ⊗ x + λa ⊗ g − λab ⊗ g which does not hold since the coefﬁcient of ab ⊗ x is 1 in the left hand sight and 0 in the right side, while if a . x = α − αg − x for some α ∈ k we get:

λa ⊗ 1 − λab ⊗ 1 + αab ⊗ 1 − αab ⊗ g − ab ⊗ x = αa ⊗ 1 − αa ⊗ g − a ⊗ x + λa ⊗ g − λab ⊗ g which again does not hold since the coefﬁcient of ab ⊗ x is −1 in the left hand sight and 0 in the right side. Thus a / g 6= ab.

178 3.5. EXAMPLES

Suppose now that a / g = b. Then we have:

a = a / g2 = (a / g) / g = b / g

Thus we get b / g = a. Also, a / x = λ(a − b) and b / x = µ(b − a) for some λ, µ ∈ k. By applying again (2.32) for a ∈ [C2 × C2] and x ∈ H4 we get:

λa ⊗ 1 − λb ⊗ 1 + b ⊗ a . x = a ⊗ a . x + λa ⊗ g − λb ⊗ g

Now observe that if a . x = x the above relation becomes:

λa ⊗ 1 − λb ⊗ 1 + b ⊗ x = a ⊗ x + λa ⊗ g − λb ⊗ g which does not hold since the coefﬁcient of b ⊗ x is 1 in the left hand side and 0 in the right hand side, while if a . x = α − αg − x for some α ∈ k we get:

λa ⊗ 1 − λb ⊗ 1 + αb ⊗ 1 − αb ⊗ g − b ⊗ x = αa ⊗ 1 − αa ⊗ g − a ⊗ x + λa ⊗ g − λb ⊗ g which again does not hold since the coefﬁcient of b ⊗ x is −1 in the left hand sight and 0 in the right side. Thus a / g 6= b. Suppose now that a / g = a and b / g = ab. Then we have: a / x = 0 and b / x = λ(b − ab) for some λ ∈ k. By applying (2.32) for b ∈ [C2 × C2] and x ∈ H4 we get:

λb ⊗ 1 − λab ⊗ 1 + ab ⊗ b . x = b ⊗ b . x + λb ⊗ g − λab ⊗ g

Now observe that if b . x = x the above relation becomes:

λb ⊗ 1 − λab ⊗ 1 + ab ⊗ x = b ⊗ x + λb ⊗ g − λab ⊗ g which does not hold since the coefﬁcient of ab ⊗ x is 1 in the left hand side and 0 in the right hand side, while if b . x = α − αg − x for some α ∈ k we get:

λb ⊗ 1 − λab ⊗ 1 + αab ⊗ 1 − αab ⊗ g − ab ⊗ x = αb ⊗ 1 − αb ⊗ g − b ⊗ x + λb ⊗ g − λab ⊗ g which again does not hold since the coefﬁcient of ab ⊗ x is −1 in the left hand sight and 0 in the right side. Thus we can not have a / g = a and b / g = ab. Finally, the only possibility left is that a / g = a and b / g = b. Then we obtain a / x = 0 and b / x = 0. Moreover, we also have:

(2.31) ab / g = a / (b . g)(b / g) = (a / g)(b / g) = ab

(2.31) ab / x = a / (b . x(1)) (b / x(2)) = a / (b . x)(b / 1) + a / (b . g)(b / x) = a / (b . x)b

179 CHAPTER 3. CLASSIFYING BICROSSED PRODUCTS OF QUANTUM GROUPS

Now if b . x = x we get ab / x = (a / x)b = 0 while if b . x = α − αg − x we obtain:

ab / x = a / (α − αg − x)b = (αa − αa − a / x)b = 0

Hence, ab / x = 0. Also, we have:

a / gx = (a / g) / x = a / x = 0 b / gx = (b / g) / x = b / x = 0 ab / gx = (ab / g) / x = ab / x = 0

This gives the trivial right action. We note that since / is the trivial action then the compatibility condition (2.30) becomes g.ab = (g(1) .a)(g(2) .b) while (2.31) and (2.32) are trivially fulﬁlled. Suppose now that a . x = x and b . x = δ − δg − x where δ ∈ k. Since

(2.31) 0 = b . x2 = (b . x)(b . x) = 2δ2 − 2δ2g − 2δx we get δ = 0 and thus b . x = −x. This corresponds to the left action given as follows:

.1 1 g x gx 1 1 g x gx a 1 g x gx b 1 g −x −gx ab 1 g −x −gx

We denote by B1 the corresponding bicrossed product. Thus B1 is deﬁned as follows: B1 =< a, b, g, x | a2 = b2 = g2 = 1, x2 = 0, ab = ba, ag = ga, bg = gb, ax = xa, xb = −bx, xg = −gx > with the coalgebra structure given by:

∆(x) = g ⊗ x + x ⊗ 1 and ∆(z) = z ⊗ z for all z ∈ {a, b, g}

Assume now that a . x = γ − γg − x, b . x = x where γ ∈ k. Since

(2.31) 0 = a . x2 = (a . x)(a . x) = (γ − γg − x)(γ − γg − x) = 2γ − 2γg − 2γx we get γ = 0 and thus a . x = −x. This corresponds to the action given as follows:

.2 1 g x gx 1 1 g x gx a 1 g −x −gx b 1 g x gx ab 1 g −x −gx

180 3.5. EXAMPLES

We denote by B2 the corresponding bicrossed product. Thus B2 is deﬁned as follows B2 =< a, b, g, x | a2 = b2 = g2 = 1, x2 = 0, ab = ba, ag = ga, bg = gb, ax = −xa, xb = bx, xg = −gx > with the coalgebra structure given by:

∆(x) = g ⊗ x + x ⊗ 1 and ∆(z) = z ⊗ z for all z ∈ {a, b, g}

Assume now that a . x = γ − γg − x and b . x = δ − δg − x where γ, δ ∈ k. As before we obtain γ = δ = 0 and thus a . x = −x, b . x = −x. This corresponds to the action given as follows:

.3 1 g x gx 1 1 g x gx a 1 g −x −gx b 1 g −x −gx ab 1 g x gx

We denote by B3 the corresponding bicrossed product. Thus B3 is deﬁned as follows: B3 =< a, b, g, x | a2 = b2 = g2 = 1, x2 = 0, ab = ba, ag = ga, bg = gb, ax = −xa, xb = −bx, xg = −gx > with the coalgebra structure given by:

∆(x) = g ⊗ x + x ⊗ 1 and ∆(z) = z ⊗ z for all z ∈ {a, b, g}

Finally, if a . x = x and b . x = x then . is the trivial action. Of course, the compatibility conditions (2.29)-(2.32) are trivially fulﬁlled and the corresponding bicrossed product is the 2 2 2 2 tensor product H4 ⊗ k[C2 × C2] =< a, b, g, x | a = b = g = 1, x = 0, ab = ba, ag = ga, bg = gb, ax = xa, xb = bx, xg = −gx > with the coalgebra structure given by:

∆(x) = g ⊗ x + x ⊗ 1 and ∆(z) = z ⊗ z for all z ∈ {a, b, g}

Next we prove that B1, B2, B3 and H4 ⊗ k[C2 × C2] are all isomorphic as Hopf algebras. To see that B1 and B2 are isomorphic we consider the map deﬁned as follows:

ψ1,2 : B1 → B2, ψ1,2(g) = g, ψ1,2(x) = x, ψ1,2(a) = b and ψ1,2(b) = a

Of course, ψ1,2 is a Hopf algebra isomorphism. In the same way we can prove that B1 is isomorphic to B3. Indeed, consider the following map:

ψ1,3 : B1 → B3, ψ1,3(g) = g, ψ1,3(x) = x, ψ1,3(a) = ab and ψ1,3(b) = b

It is straightforward to see that ψ1,3 is also a Hopf algebra isomorphism.

Finally, to prove that B1 is isomorphic to H4 ⊗ k[C2 ⊗ C2] we consider the following Hopf algebra isomorphism ψ : B1 → H4 ⊗ k[C2 × C2] given as follows:

ψ(g) = g, ψ(x) = x, ψ(a) = a and ψ(b) = gb

181 CHAPTER 3. CLASSIFYING BICROSSED PRODUCTS OF QUANTUM GROUPS

The results proven in Proposition 3.5.3 (resp. Proposition 3.5.15) show that the left action / of any matched pair (H4, k[Cn], ., /) (resp. (H4, k[C2 × C2], ., /)) is necessarily the trivial action. Hence any bicrossed product H4 ./ k[Cn] (resp. H4 ./ k[C2 × C2]) is in fact isomorphic to a semi-direct (smash) product H4#k[Cn] (resp. H4#k[C2 × C2]). These can be viewed as Schur- Zassenhaus type theorems for the bicrossed product of Hopf algebras. [11, Theorem 2.1] proves a similar result at the level of groups: any bicrossed product of two finite cyclic groups, one of them being of prime order, is isomorphic to a semidirect product between the two cyclic groups of the same order. A matched pair (A, H, ., /) of Hopf algebras is called proper if / and . are both nontrivial actions. The next aim is to give an example of a classification result of matched pairs of Hopf algebras (A, H, ., /) among which there exist proper matched pairs of Hopf algebras. Having in mind the Drinfel’d double case from Example 3.1.2, these examples exist if we look to classify all matched pairs between (H∗)cop and H, where H is a finite dimensional noncommutative noncocommutative Hopf algebra. Among them there will certainly be the matched pair given by (3.6), with the associated bicrossed product D(H) = (H∗)cop ./ H, the Drinfel’d double of H. The interesting part of the story is that D(H) might not be the only one: besides D(H) there might be other finite quantum groups which are non-trivial bicrossed products between (H∗)cop and H. Sweedler’s Hopf algebra will provide such an example. Being, up to an isomorphism, the only 4-dimensional, noncommutative and nococommutative Hopf algebra, we have ∗ cop ∼ that (H4 ) = H4, an isomorphism of Hopf algebras. To avoid confusions we will denote by 2 H4 a copy of H4, i.e. H4 is the k-algebra generated by G and X subject to the relations G = 1, X2 = 0, GX = −XG, and with the coalgebra structure such that G is a group-like element and X is an (1,G)-primitive element. The next result, proved in [36], describes by generators and relations and classifies all Hopf algebras that factorize through two Sweedler’s Hopf algebras.

Theorem 3.5.16 Let k be a ﬁeld of characteristic 6= 2. Then: ∼ ∼ (1) A Hopf algebra E factorizes through H4 and H4 if and only if E = H4 ⊗ H4 or E = H16, λ, for some λ ∈ k, where H16, λ is the Hopf algebra generated by g, x, G, X subject to the relations: g2 = G2 = 1 x2 = X2 = 0, gx = −xg, GX = −XG, gG = Gg, gX = −Xg, xG = −Gx, xX + Xx = λ (1 − Gg) with the coalgebra structure given by ∆(g) = g ⊗ g, ∆(x) = x ⊗ 1 + g ⊗ x, ∆(G) = G ⊗ G, ∆(X) = X ⊗ 1 + G ⊗ X, ε(g) = ε(G) = 1, ε(x) = ε(X) = 0

(2) H16, λ is a pointed non-semisimple 16-dimensional Hopf algebra and the canonical extension H4 ⊆ H16, λ is rigid. (3) Up to an isomorphism of Hopf algebras, there are only three Hopf algebras that factorize through H4 and H4, namely ∼ H4 ⊗ H4, H16, 0 and H16, 1 = D(H4)

182 3.5. EXAMPLES

where D(H4) is the Drinfel’d double of H4.

Remark 3.5.17 At this point we should notice that H16,0 is a new Hopf algebra and not just the ∗ dual of D(H4). This can be seen by looking at the primitive elements of H16,0. By a straightfor- ∗ ∗ ∗ ∗ ∗ ward computationit follows that the group-like elements of H16,0 are 1 + g + G + (Gg) = ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 1 ∗ , 1 −g −G +(Gg) , 1 +g −G −(Gg) and 1 −g +G −(Gg) and, for any pair or H16,0 ∗ 2 group-like elements different from 1 ∗ , (g , g ), one has P ∩ {h ∈ H | h = 0} = {0} H16,0 1 2 1,g1 16,0 ∗ 2 or P1,g2 ∩ {h ∈ H16,0 | h = 0} = {0}. Since in D(H4), g and G are group-like elements 2 different from 1 such that 0 6= x ∈ P1,g ∩ {h ∈ D(H4) | h = 0} and 0 6= X ∈ P1,G ∩ {h ∈ 2 D(H4) | h = 0}, we can conclude that H16,0 is not the dual of D(H4).

183 CHAPTER 3. CLASSIFYING BICROSSED PRODUCTS OF QUANTUM GROUPS

3.6 Application: Bicrossed descent theory for groups

All results proved in Section 3.4 can be particularized to the group case. Let G be a group, A ≤ G be a subgroup of G and consider the Hopf algebra extension k[A] ⊆ k[G], where E := k[G] is the group algebra of G. The following are just elementary facts: any Hopf subalgebra of a group algebra k[G] is of the form k[H], for a subgroup H ≤ G; Hopf algebra morphisms between two group algebras k[H] and k[A] are in bijective correspondence with morphisms of groups between H and A; coalgebra maps between k[H] and k[A] are in bijection with usual functions between the sets H and A; the set of all matched pairs of Hopf algebras (k[A], k[H], ., /) is in bijection with the set of all matched pairs of groups (A, H, ., /) and the bijection is given such that k[A] ./ k[H] =∼ k[A ./ H] (isomorphism of Hopf algebras), where A ./ H is the bicrossed product of groups in the sense of [143]; the group algebra E := k[G] factorizes as a Hopf algebra through two Hopf subalgebras is and only if the group G factorizes through two subgroups. Using these observations all the results proved above can be translated to groups without needing a proof. We only write down brieﬂy the new concepts and results that are obtained. We recall the following well-known fact:

Proposition 3.6.1 A group G factorizes through two subgroups A and H if and only if there exists a matched pair of groups (A, H, ., /) such that the multiplication map

mG : A ./ H → G, mG(a, h) = ah for all a ∈ A and h ∈ H is an isomorphism of groups. Furthermore, in this case the isomorphism of groups mG : A ./ H → G stabilizes A.

Proof: The detailed proof is given in [143, Proposition 2.4] or [81, Proposition IX.1.2]. We only indicate the construction of the matched pair (A, H, ., /) associated to the exact factorization G = AH. Indeed, if G factorizes through A and H then, for any g ∈ G there exists a unique pair (a, h) ∈ A × H such that g = ah. This allows us to attach to any pair (a, h) ∈ A × H a unique element h . a ∈ A and a unique element h / a ∈ H such that

h a = (h . a)(h / a) (3.92)

Then (A, H, ., /) is a matched pair of groups.

Form now on, the matched pair constructed in (3.92) will be called the canonical matched pair associated to the factorization G = AH. Let A ≤ G be a subgroup of G.A factorization A-form of G is a subgroup H ≤ G such that G factorizes through A and H, i.e. G = AH and A ∩ H = {1} (or equivalently, there exists an isomorphism of groups G =∼ A ./ H). We denote by F(A, G) the category of all factorization A-forms of G and by F sk(A, G) its skeleton. The factorization index of A in G is deﬁned as

[G : A]f = | F sk(A, G) |

184 3.6. APPLICATION: BICROSSED DESCENT THEORY FOR GROUPS

The extension A ≤ G is called rigid if [G : A]f = 1. We shall write [G : A]f = 0, if F(A, G) is empty.

Let Sn be the symmetric group on n letters: Sn−1 will be viewed as a subgroup of Sn by letting n to be a ﬁxed point. We denote by Cn the cyclic group of order n: it will be viewed as a subgroup of Sn, being generated by the cycle (12 ··· n).

Examples 3.6.2 1. Most of the group extensions A ≤ G known in the literature have the factorization index [G : A]f equal to 0 (that is there exists no factorization G = AH) or 1. For instance, if G is an indecomposable group, then [G : A]f = 0, for any proper subgroup A < G. Group extensions A ≤ G of factorization index 1 are exactly those for which the factorization is unique. There are plenty of such examples in the theory of factorizations of ﬁnite simple groups: see for instance the tables in [68], [88], [89] that classify all factorizations of a given simple group G. 2. Examples of extensions G/A for which [G : A]f ≥ 2 are quite rare, which makes them tempting to identify. We provide below two examples of extensions of factorization index 2 and 3.

The extension S3 ≤ S4 has factorization index 2. Indeed, let C4 =< (1234) > be the cyclic group of order 4 and C2 × C2 the Klein’s group viewed as a subgroup of S4 being generated by (12)(34) and (13)(24). Then S4 has two factorizations: S4 = S3C4 = S3(C2 × C2). Since f there are no other groups of order four we obtain that [S4 : S3] = 2. + 3. Let A = A9 and G be the group of Lie type G = Ω8 (2). It was proven in [89, Chapter 16, + pg.90], using Magma, that G has precisely three non-isomorphic factorizations. Thus [Ω8 (2) : f A9] = 3. 4. Example (2) above can be generalized to a broader case as follows: the factorization index f [Sn : Sn−1] = GR (n), the number of types of isomorphisms of groups of order n. ∼ Indeed, let Sn be the symmetric group and A = Sn−1 the stabilizer of a point. Then we have an exact factorization Sn = AH, for a regular subgroup H ≤ Sn of order n. Since any group H of order n acts transitively on itself by right multiplication it embeds in Sn as a transitive subgroup and Sn = Sn−1H, where Sn−1 is any copy of Sn−1 in Sn as the stabilizer of a point.

Let (A, H, ., /) be a given matched pair of groups. A (., /)-cocycle is a function r : H → A such that r(1H ) = 1A and for any g, h ∈ H we have: r h / r(g) g = r(h) h . r(g)

We denote by HA1(H,A | (., /)) the set of all (., /)-cocycles. Let r : H → A be a (., /)- cocycle. Then it follows from Theorem 3.4.7 that Hr := H (as a set) has a new group structure with the multiplication given by h • g := h / r(g) g for all h, g ∈ H called the r-deformation of the group H. Now, Theorem 3.4.9 and Theo- rem 3.4.10 provide the following result:

185 CHAPTER 3. CLASSIFYING BICROSSED PRODUCTS OF QUANTUM GROUPS

Theorem 3.6.3 Let A ≤ G be a subgroup of G and H a given factorization A-form of G with the associated matched pair (A, H, ., /). Then H is a factorization A-form of G if and only if ∼ there exists an isomorphism of groups H = Hr, for some (., /)-cocycle r : H → A, where Hr is the r-deformation of H.

Remark 3.6.4 Assume that in Theorem 3.4.7 the descent map r : H → A is the trivial one or the right action / is the trivial action of A on H. Then Hr = H as groups. In general, the new group Hr may not be isomorphic to H as groups. Example 3.6.7 shows how the Klein’s group C2 × C2 can be constructed as an r-deformation of the cyclic group C4, for some descent map r : C4 → S3. On the other hand, there are also examples of non-trivial descent maps, with / a non-trivial action, such that Hr is a group isomorphic to H. Such an example is provided in Example 3.6.9.

Two (., /)-cocycles r, R : H → A are called equivalent and we denote this by r ∼ R if there exists σ : H → H a permutation on the set H such that σ(1H ) = 1H and

σ(h / r(g)) g= σ(h) /R(σ(g)) σ(g) for all g, h ∈ H. Theorem 3.4.6 takes the form:

Theorem 3.6.5 Let A ≤ G be a subgroup of G and H a given factorization A-form of G with the associated matched pair (A, H, ., /). Then ∼ is an equivalence relation on HA1(H,A | (., /)) and the map

2 1 sk HA (H,A | (., /)) := HA (H,A | (., /))/ ∼ → F (A, E), r 7→ Hr is well deﬁned and a bijection between sets, where r is the equivalence class of r via the relation ∼. In particular, the factorization index [G : A]f is computed by the formula:

[G : A]f = |HA2(H,A | (., /))|

The main consequence of the bicrossed descent theory is the following:

Corollary 3.6.6 Let n be a positive integer and (Sn−1,Cn, ., /) the canonical matched pair associated to the factorization Sn = Sn−1Cn. Then:

(1) Any group H of order n is isomorphic to a r-deformation of the cyclic group Cn, for some descent map r : Cn → Sn−1 of the canonical matched pair (Sn−1,Cn, ., /). (2) The number of types of isomorphisms of all groups of order n is equal to

| D(Cn,Sn−1 | (., /)) |

Proof: It follows from Theorem 3.6.3 and Theorem 3.6.5 taking into account that any group H of order n is a Sn−1-form of Sn according to (4) of Example 3.6.2.

186 3.6. APPLICATION: BICROSSED DESCENT THEORY FOR GROUPS

Now we provide some explicit examples for the above results.

Example 3.6.7 Consider the extension S3 ≤ S4 of factorization index 2 from Example 3.6.2. We ﬁx C4 as a S3-form of S4. First we shall describe the canonical matched pair (S3,C4, ., /) associated to the factorization S4 = S3C4. Then we prove that there exists only two descent maps r : C4 → S3 for the matched pair (S3,C4, ., /). Finally, we show how the Klein’s group is written as an r-deformation of the cyclic group C4. 4 Below we shall see C4 and C2 × C2 as subgroups of S4 as follows: C4 = hc = (1234) | c = 1i 2 2 and C2 × C2 = ha = (12)(34), b = (13)(24) | a = b = 1, ab = bai. We also use the 3 2 2 description of S3 as: S3 = ht = (123), s = (12) | t = s = 1, ts = st i. Using these notations, a straightforward computation proves that the canonical matched pair (S3,C4, ., /) associated as in Proposition 3.6.1 to the factorization S4 = S3C4 is given by: . 1 s t t2 st st2 / 1 s t t2 st st2 1 1 s t t2 st st2 1 1 1 1 1 1 1 c 1 st s t t2 st2 c c c c2 c3 c2 c3 c2 1 t2 st s t st2 c2 c2 c3 c3 c c c2 c3 1 t t2 st s st2 c3 c3 c2 c c2 c3 c

By a rather long but straightforward computation one can prove that there are two descent maps 0 0 for the canonical matched pair (S3,C4, ., /): namely the trivial one r : C4 → S3, r (x) = 1, for any x ∈ C4 and the map given by 2 3 2 r : C4 → S3, r(1) = r(c ) = 1, r(c) = r(c ) = st

Now, the multiplication • on (C4)r given by (3.65) takes the following form:

• 1 c c2 c3 1 1 c c2 c3 c c 1 c3 c2 c2 c2 c3 1 c c3 c3 c2 c 1

Thus, the map given by: 2 3 ϕ : C2 × C2 → (C4)r, ϕ(1) = 1, ϕ(a) = c, ϕ(b) = c , ϕ(ab) = c ∼ is an isomorphism of groups, that is C2 × C2 = (C4)r.

Corollary 3.6.6 proves that any ﬁnite group of order n is isomorphic to a r-deformation of the cyclic group Cn, for some descent map r : Cn → Sn−1 of the canonical matched pair associated to the factorization Sn = Sn−1Cn. The next example shows how the symmetric group S3 appears as a r-deformation of the cyclic group C6 arising from a given matched pair (C3,C6, ., /).

3 6 Example 3.6.8 In what follows we denote by C3 = ha | a = 1i and C6 = hb | b = 1i the cyclic group of order 3 respectively 6. As a special case of [13, Proposition 4.2] we have a matched pair (C3,C6, ., /), where the actions (., /) are given by:

187 CHAPTER 3. CLASSIFYING BICROSSED PRODUCTS OF QUANTUM GROUPS

. 1 a a2 / 1 a a2 1 1 a a2 1 1 1 1 b 1 a2 a b b b3 b5 b2 1 a a2 b2 b2 b2 b2 b3 1 a2 a b3 b3 b5 b b4 1 a a2 b4 b4 b4 b4 b5 1 a2 a b5 b5 b b3

Now, by a rather long but straightforward computation it can be seen that the map

3 4 2 2 5 r : C6 → C3, r(1) = r(b ) = 1, r(b) = r(b ) = a , r(b ) = r(b ) = a is a descent map of the matched pair (C3,C6, ., /). Furthermore, the multiplication • on (C6)r given by (3.65) takes the form:

• 1 b b2 b3 b4 b5 1 1 b b2 b3 b4 b5 b b 1 b5 b4 b3 b2 b2 b2 b3 b4 b5 1 b b3 b3 b2 b 1 b5 b4 b4 b4 b5 1 b b2 b3 b5 b5 b4 b3 b2 b 1

Thus, the map

2 2 4 5 2 3 ϕ : S3 → (C6)r, ϕ(1) = 1, ϕ(s) = b, ϕ(r) = b , ϕ(r ) = b , ϕ(sr) = b , ϕ(sr ) = b is an isomorphism of groups and hence S3 is an r-deformation of the cyclic group C6.

∼ Our last example provides a non-trivial descent map r : H → A such that Hr = H.

Example 3.6.9 Let (C3,C6, ., /) be the matched pair of Example 3.6.8. Then the map

2 4 3 5 R : C6 → C3,R(1) = R(b ) = R(b ) = 1,R(b) = R(b ) = R(b ) = a is also a descent map of (C3,C6, ., /). Then, one can check by a straightforward computation that (C6)R is a group isomorphic to C6.

Bibliographical Notes

All material in this chapter originates from the author’s joint work with G. Bontea and G. Militaru [8] except for Section 3.6 which is inspired on the author’s joint work with G. Militaru [17] and Section 3.4 which is taken from the author’s joint paper with G. Militaru [14].

188 Bibliography

[1] Abe, E. - Hopf Algebras, Cambridge University Press, 1977.

[2] Adamek,´ J., Rosicky,´ J. - Locally Presentable and Accessible Categories, Cambridge University Press, 1994.

[3] Agore, A.L. - Limits of coalgebras, bialgebras and Hopf algebras, Proc. Amer. Math. Soc. 139 (2011), 855 – 863.

[4] Agore, A.L. - Categorical constructions for Hopf algebras, Comm. Algebra 39 (2011), 1476 – 1481.

[5] Agore, A.L. - Monomorphisms of coalgebras, Colloq. Math. 120 (2010), 149 – 155.

[6] Agore, A.L. - Coquasitriangular structures for extensions of Hopf algebras, Glasgow Math. J. 55 (2013), 201 – 215.

[7] Agore, A.L. - Crossed product of Hopf algebras, Comm. Algebra 40 (2013), 2519 – 2542.

[8] Agore, A.L., Bontea, G., Militaru, G. - Classifying bicrossed products of Hopf algebras, Algebr. Represent. Theory 17 (2014), 227–264.

[9] Agore, A.L., Caenepeel, S., Militaru, G. - Braidings on the category of bimodules, Azu- maya algebras and epimorphisms of rings, Appl. Cat. Structures 22 (2014), 29–42.

[10] Agore, A.L., Caenepeel, S., Militaru, G. - The center of the category of bimodules and descent data for non-commutative rings, 14 pages, J. Algebra Appl. 11 (2012), 1–17.

[11] Agore, A.L., Chirvasitu, A., Ion, B., Militaru, G. - Bicrossed products for ﬁnite groups, Algebr. Represent. Theory 12 (2009), 481 – 488.

189 BIBLIOGRAPHY

[12] Agore, A.L., Militaru, G. - Extending structures II: the quantum version, J. Algebra 336 (2011), 321 – 341.

[13] Agore, A.L., Militaru, G. - Schreier type theorems for bicrossed products, Cent. Eur. J. Math. 10 (2012), 722 – 739.

[14] Agore, A.L., Militaru, G. - Classifying complements for Hopf algebras and Lie algebras, J. Algebra 391 (2013), 193–208.

[15] Agore, A.L., Militaru, G. - Uniﬁed products and split extensions of Hopf algebras, AMS Contemporary Math. 585 (2013), 1–15.

[16] Agore, A.L., Militaru, G. - Extending structures I: the level of groups, Algebr. Represent. Theory 17 (2014), 831–848.

[17] Agore, A.L., Militaru, G. - Classifying complements for groups. Applications, 13 pages, 13 pages, to appear in Ann. Inst. Fourier.

[18] Aguiar, M., Andruskiewitsch, N., - Representations of matched pairs of groupoids and applications to weak Hopf algebras. Algebraic structures and their representations, AMS Contemporary Math. 376 (2005), 127 – 173.

[19] Alperin, J.L., Bell, R.B. - Groups and representations, Graduate Texts in Mathematics 162, Springer 1995.

[20] Amberg, B., Franciosi, S., Giovanni, F. - Products of groups, Oxford University Press, 1992.

[21] Andruskiewitsch, N., Devoto, J. - Extensions of Hopf algebras, Algebra i Analiz 7 (1995), 22 – 61.

[22] Andruskiewitsch, N., Schneider, H.-J. - Lifting of Quantum Linear Spaces and Pointed Hopf Algebras of order p3, J. Algebra 209 (1998), 658 – 691.

[23] Andruskiewitsch, N., Schneider, H.-J. - On the classiﬁcation of ﬁnite-dimensional pointed Hopf algebras, Ann. Math. 171 (2010), 375 - 417.

[24] Ardizzoni, A., Beattie, M., Menini, C. - Cocycle deformations for Hopf algebras with a coalgebra projection, J. Algebra 324 (2010), 673 – 705.

[25] Ardizzoni, A., Menini, C. - Small Bialgebras with a Projection: Applications, Comm. Algebra 37 (2009), 2742 – 2784.

[26] Ardizzoni, A., Menini, C., Stefan, D. - A monoidal approach to splitting morphisms of bialgebras, Trans. Amer. Math. Soc. 359 (2007), 991 – 1044.

[27] Ardizzoni, A., Menini, C., Stumbo, F. - Small Bialgebras with Projection, J. Algebra 314 (2007), 613 – 663.

190 BIBLIOGRAPHY

[28] Auslander, M., Goldman, O. - The Brauer group of a commutative ring, Trans. Amer. Math. Soc. 97 (1960), 367 – 409.

[29] Azumaya, G. - On maximally central algebras, Nagoya Math. J. 2 (1951), 119 – 150.

[30] Baaj, S., Skandalis, G., Vaes, S. - Measurable Kac cohomology for bicrossed products, Trans. Amer. Math. Soc. 357 (2005), 1497 – 1524.

[31] Baumeister, B. - Factorizations of Primitive Permutation Groups, J. Algebra 194 (1997), 631 – 653.

[32] Bespalov, Y., Drabant B. - Crossed products of bialgebras I, J. Algebra 219 (1999), 466 – 505.

[33] Bespalov, Y., Drabant B. - Crossed products of bialgebras II, J. Algebra 240 (2001), 445 – 504.

[34] Bichon, J., Kassel, C. - The lazy homology of a Hopf algebra, J. Algebra 323 (2010), 2556 – 2590.

[35] Blattner, R.J., Cohen, M., Montgomery, S. - Crossed product and inner actions of Hopf algebras, Trans. Amer. Math. Soc. 298 (1986), 671 – 711.

[36] Bontea, C.G. - Classifying bicrossed products of two Sweedler’s Hopf algebras, 22 pages, preprint 2012, arXiv:1205.7010.

[37]B ohm,¨ G. - Hopf algebroids, Handbook Algebra 6 (2009), 173 – 235.

[38]B ohm,¨ G., Caenepeel, S., Janssen, K. - Weak bialgebras and monoidal categories, Comm. Algebra 39 (2011), 4584 – 4607.

[39]B ohm,¨ G., Nill, F., Szlachanyi,´ K. - Weak Hopf algebras I. Integral theory and C∗- structure, J. Algebra 221 (1999), 385 – 438.

[40] Brzezinski,´ T. - The structure of corings. Induction functors, Maschke-type theorem, and Frobenius and Galois properties, Algebr. Represent. Theory 5 (2002), 389 – 410.

[41] Brzezinski,´ T. - Flat connections and (co)modules, in “New Techniques in Hopf Algebras and Graded Ring Theory”, S. Caenepeel and F. Van Oystaeyen (eds), Universa Press, Wetteren, 2007, 35 – 52.

[42] Brzezinski,´ T. - A note on ﬂat noncommutative connections, 10 pages, preprint 2011, arXiv:1109.0858.

[43] Brzezinski,´ T., Wisbauer, R. - Corings and comodules, London Math. Soc. Lect. Note Ser. 309, Cambridge University Press, 2003.

[44] Burciu, S. - On complements and the factorization problem for Hopf algebras, Cent. Eur. J. Math. 9 (2011), 905 – 914.

191 BIBLIOGRAPHY

[45] Caenepeel, S. - Brauer groups, Hopf algebras and Galois theory, K-Monographs in Math- ematics 4, Kluwer Academic Publishers, Dordrecht, 1998.

[46] Caenepeel, S. - Galois corings from the descent theory point of view, Fields Inst. Comm. 43 (2004), 163 – 186.

[47] Caenepeel, S., Militaru, G., Zhu, S. - Frobenius and separable functors for generalized module categories and nonlinear equations, Lecture Notes in Math. 1787, Springer Ver- lag, Berlin, 2002.

[48] Caenepeel, S., Ion, B., Militaru, G., Zhu, S. - The factorization problem and the smash biproduct of algebras and coalgebras. Algebr. Represent. Theory 3 (2000), 19 – 42.

[49] Cap, A., Schichl, H., Vanzura, J. - On twisted tensor product of algebras, Comm. Algebra 23 , 4701 – 4735.

[50] Castano˜ Iglesias, F., Dascalescu, S., Nastasescu, C. - Symmetric coalgebras, J. Algebra 279 (2004), 326 – 344.

[51] Chandrasekharan, K. - Introduction to Analytic Number Theory, Graduate Texts in Mathematics 148, Springer 1968.

[52] Chin, W. - Crossed products of semisimple cocomutative Hopf algebras, Proc. Amer. Math. Soc. 116 (1992), 321 – 327.

[53] Chirvsitu, A. - On epimorphisms and monomorphisms of Hopf algebras, J. Algebra 323 (2010), 1593 – 1606.

[54] Cipolla, M. - Discesa fedelmente piatta dei moduli, Rend. Circ. Mat. Palermo (2) 25 (1976), 43 – 46.

[55] Cirio, L.S., Pagani, C. - Deformation of tensor product (co)algebras via non-(co)normal twists, 25 pages, preprint 2011, arXiv:1112.2992.

[56] Cohn, P.M. - A remark on the general product of two inﬁnite Cyclic groups, Arch. Math. (Basel) 7 (1956), 94 – 99.

[57] Darafsheh, M.R., - Finite groups which factor as product of an alternating group and symmetric group, Comm. Algebra 32 (2004), 637 – 647.

[58]D ascˇ alescu,ˇ S., Nastˇ asescu,ˇ C., Raianu, S. - Hopf algebras: An introduction, Pure and Applied Math. 235, Marcel Dekker, 2000.

[59] DeMeyer, F., Janusz, G. J. - Group rings which are Azumaya algebras, Trans. Amer. Math. Soc. 279 (1983), 389 – 395.

[60] Doi, Y. - Braided bialgebras and quadric bialgebras, Comm. Algebra 21(1993), 1731 – 1749.

192 BIBLIOGRAPHY

[61] Doi, Y. - Equivalent crossed products for a Hopf algebra, Comm. Algebra 17 (1989), 3053 – 3085.

[62] Doi, Y., Takuchi, M. - Cleft comodule algebras for a bialgebras, Comm. Algebra 14 (1986), 801 – 818.

[63] Doi, Y., Takeuchi, M. - Multiplication alteration by two-cocycles, Comm. Algebra 22 (1994), 5715 – 5732.

[64] Donin, J., Mudrov, A. - Quantum groupoids and dynamical categories J. Algebra 296 (2006), 348 – 384.

[65] Douglas, J. - On ﬁnite groups with two independent generators. I, II, III, IV. Proc. Nat. Acad. Sci. U. S. A. 37 (1951), 604 – 610, 677 – 691, 749 – 760, 808 – 813.

[66] Drinfeld, V. - Quantum groups, Proc. Intern. Congr. Math., Berkeley, Vol. I (1987), 798 – 820.

[67] Eilenberg, S., MacLane, S., - Cohomology theory in abstract groups I, II, Ann. Math. 48 (1947), 51 - 78 and 326 - 341.

[68] Giudici, M. - Factorisations of sporadic simple groups, J. Algebra 304 (2006), 311 – 323.

[69] Grothendieck, A. - Technique de Descente I, Sem.´ Bourbaki, exp. 190 (1959-1960).

[70] Guccione, J. A., Guccione, J.J., Valqui, C., - Twisted planes, Comm. Algebra 38 (2010), 1930 – 1956.

[71] Haggenmuller,¨ R., Pareigis, B. - Hopf algebra forms of the multiplicative group and other groups, Manuscripta Math. 55 (1986), 121 – 136.

[72] Hilgemann, M., Ng, S.-H. - Hopf algebras of dimension 2p2, J. London Math. Soc. 80 (2009), 295 – 310.

[73]H older,¨ O. - Bildung zusammengesetzter Gruppen, Math. Ann. 46 (1895), 321 – 422.

[74] Hungerford, T. W. - Algebra, Graduate Texts in Mathematics 73, Springer, 1974.

[75] Ito, N. - ber das Produkt von zwei abelschen Gruppen, Math. Z. 62 (1955), 400 – 401.

[76] Jara Martinez, P., Lopez´ Pena,˜ J., Panaite, F. and Van Oystaeyen, F. - On Iterated Twisted Tensor Products of Algebras, Internat. J. Math. 19 (2008), 1053 – 1101.

[77] Jiao, Z., Wisbauer, R. - The braided structures for ω-smash coproduct Hopf algebras, J. Algebra 287 (2005), 474 – 495.

[78] Jiao, Z., Wisbauer, R. - The braided structures for T -smash product Hopf algebras, In- ternational Electronical Journal of Algebra 1 (2007), 30 – 45.

193 BIBLIOGRAPHY

[79] Joyal, A., Street, R. - Braided tensor categories, Adv. Math. 102 (1993), 20 – 78.

[80] Joyal, A., Street, R. - Tortile Yang-Baxter operators in tensor categories, J. Pure Appl. Algebra 71 (1991), 43 – 51.

[81] Kassel, C. - Quantum groups, Graduate Texts in Mathematics 155, Springer-Verlag, New York, 1995.

[82] Knight, J.T. - On epimorphisms of non-commutative rings, Proc. Cambridge Phil. Soc. 68 (1970), 589 – 600.

[83] Knus, M. A., Ojanguren, M., - Theorie´ de la descente et algebres` d’Azumaya, Lecture Notes in Math. 389, Springer Verlag, Berlin, 1974.

[84] Kroetz, B. - A novel characterization of the Iwasawa decomposition of a simple Lie group, Springer Lecture Notes in Physics 723 (2007), 195 – 201.

[85] Kuperberg, G. - Exact factorization of finite groups, Available online at: http://mathoverflow.net/questions/55110/exact-factorization-of-finite-groups

[86] Larson, R., Radford, D. - Finite-dimensional cosemisimple Hopf algebras in characteristic 0 are semisimple, J. Algebra 117 (1988), 267 – 289.

[87] Larson, R., Towber, J. - Two dual classes of bialgebras related to the concepts of ”quantum groups” and ”quantum Lie algebras”, Comm. Algebra 19 (1991), 3295 – 3345.

[88] Liebeck, M.W., Praeger, C. E., Saxl, J. - The maximal factorizations of the ﬁnite simple groups and their automorphism groups, Mem. Amer. Math. Soc. 86 (1990).

[89] Liebeck, M.W., Praeger, C. E., Saxl, J. - Regular subgroups of primitive permutation groups, Mem. Amer. Math. Soc. 203 (2010).

[90]L opez´ Pena,˜ J. and Navarro, G. - On the classiﬁcation and properties of noncommutative duplicates. K–Theory 38 (2008), 223 – 234.

[91]L opez´ Pena,˜ J. and Stefan, D., - On the classiﬁcation of twisting maps between Kn and Km, Algebr. Represent. Theory 14 (2011), 869 – 895.

[92] Ma, T.; Li, H. and Zhao, W. - On the braided structures of Radford’s biproduct, Acta Math. Sci. Ser. B Engl. 31 (2011), 701 – 715.

[93] Mac Lane, S. - Categories for the working mathematician, Graduate Texts in Mathemat- ics 5, Springer, 1998.

[94] Maillet, E. - Sur les groupes echangeables´ et les groupes decomposables.´ Bull. Soc. Math. France 28 (1900), 7 – 16.

[95] Majard, D. - On Double Groups and the Poincare group, 36 pages, preprint 2012, arXiv:1112.6208.

194 BIBLIOGRAPHY

[96] Majid, S. - Matched Pairs of Lie Groups and Hopf Algebra Bicrossproducts, Nuclear Physics B, Proc. Supl. 6 (1989), 422 – 424.

[97] Majid, S. - Quantum double for quasi-Hopf algebras, Lett. Math. Phys. 45 (1998), 1 – 9.

[98] Majid, S. - Representations, duals and quantum doubles of monoidal categories, Rend. Circ. Math. Palermo (2) 26 (1991), 197 – 206.

[99] Majid, S., Soibelman, Ya. S. - Bicrossproduct structure of the quantum Weyl group, J. Algebra 163 (1994), 68 – 87.

[100] Majid, S. - Physics for algebraists: non-commutative and non-cocommutative Hopf algebras by a bicrossproduc construction, J. Algebra 130 (1990), 17 – 64.

[101] Majid, S. - Foundations of quantum groups theory, Cambridge University Press, 1995.

[102] Majid, S. - Quantum groups and quantum probability, in Quantum Probability and Re- lated Topics VI (Proc. Trento 1989), World Science 1991.

[103] Michor, P. W. - Knit products of graded Lie algebras and groups, Proceedings of the Win- ter School on Geometry and Physics (Srn, 1989). Rend. Circ. Mat. Palermo 22 (1990), 171 – 175.

[104] Milies, C. P., Sehgal, S. K. - An introduction to group rings, Kluwer Academic Publish- ers, Dordrecht, 2002.

[105] Molnar, R.K. - Semi-direct products of Hopf algebras, J. Algebra 47 (1977), 29 – 51.

[106] Montgomery, S. - Hopf algebras and their actions on rings, CBMS Regional Conference 82, AMS, 1993.

[107] Nakagami, Y., Takesaki, M. - Duality for crossed products of von Neumann algebras, Lecture Notes in Math. 731, Springer-Verlag, Berlin, 1979.

[108]N astˇ asescu,ˇ C., Torrecillas, B. - Torsion theories for coalgebras, J. Pure and Appl. Alge- bra 97 (1994), 203 – 220.

[109]N astˇ asescu,ˇ C., Van den Bergh, M., Van Oystaeyen, F. - Separable functors applied to graded rings, J. Algebra 123 (1989), 397–413.

[110] Ng, S.-H. - Hopf algebras of dimension 2p, Proc. Amer. Math. Soc. 133 2005, 2237 – 2242.

[111] Nill, F. - Axioms for weak bialgebras, 48 pages, preprint 1998, arXiv:math/9805104.

[112] Nuss, P., Wambst, M. - Non-abelian Hopf cohomology, J. Algebra 312 (2007), 733 – 754.

[113] Nuss, P., Wambst, M. - Non-abelian Hopf cohomology II - The general case, J. Algebra 319 (2008), 4621 – 4645.

195 BIBLIOGRAPHY

[114] Oinert,¨ J. , Silvestrov, S. - Commutativity and ideals in algebraic crossed products, Jour- nal of Generalized Lie Theory and Applications 2 (2008), 287 – 302.

[115] Ore, O. - Structures and group theory. I. Duke Math. J. 3 (1937), 149 – 174.

[116] Panaite, F. - Equivalent crossed products and cross product bialgebras, preprint 2012, arXiv:1203.0037.

[117] Pareigis, B. - Twisted group rings, Comm. Algebra 17 (1989), 2923 – 2939.

[118] Pareigis, B. - Lectures on quantum groups and noncommutative geometry. Available at: http://www.mathematik.uni-muenchen.de/ pareigis/Vorlesungen/02SS/QGandNCG.pdf

[119] Pareigis, B. - Advanced Algebra. Available at: http://www.mathematik.uni-muenchen.de/ pareigis/Vorlesungen/01WS/advalg.pdf

[120] Pareigis, B., Sweedler, M.E. - On generators and cogenerators, Manuscripta Math. 2 (1970), 49 – 66.

[121] Parker, D.B., - Forms of previous Coalgebras and Hopf algebras, J. Algebra 239 (2001), 1 – 34.

[122] Passman, D. S. - Group rings, crossed products and Galois Theory, CBMS Regional Conference 64, AMS, 1986.

[123] Porst, H.-E. - Universal constructions for Hopf algebras, J. Pure Appl. Algebra 212 (2008), 2547 – 2554.

[124] Porst, H.-E. - Fundamental constructions for coalgebras, corings and comodules, Appl. Categor. Struct. 16 (2008), 223 – 238.

[125] Porst, H.-E. - On corings and comodules, Arch. Math. (Brno) 42 (2006), 419 – 425.

[126] Praeger, C. E., Schneider, C. - Factorisations of characteristically simple groups, J. Al- gebra 255 (2002), 198 – 220.

[127] Quinn, D., Raianu, S. - Semiprime crossed products over copointed Hopf algebras, Proc. Amer. Math. Soc. 131 (2003), 29 – 33.

[128] Radford, D. E. - The Structure of Hopf Algebras with a Projection, J. Algebra 92 (1985), 322 – 347.

[129] Radford, D.E. - On Kauffmans Knot Invariants Arising from Finite-Dimensional Hopf algebras, Advances in Hopf algebras, Lectures notes in pure and applied mathematics, Marcel Dekker 158, 205 – 266, 1994.

[130] Radford, D.E., Taft, E.J., Wilson, R.L. - Forms of certain Hopf algebras, Manuscripta Math. 17 (1975), 333 – 338.

[131] Rafael, M. D. - Separable functors revisited, Comm. in Algebra 18 (1990), 1445 – 1459.

196 BIBLIOGRAPHY

[132]R edei,´ L. - Zur Theorie der faktorisierbaren Gruppen I, Acta Math. Acad. Sci. Hung. 1 (1950), 74 – 98.

[133] Rotman, J. - An introduction to the theory of groups. Fourth edition. Graduate Texts in Mathematics 148, Springer-Verlag, New York, 1995.

[134] Schauenburg, P. - Duals and doubles of quantum groupoids (×R-Hopf algebras), AMS Contemporary Math. 267 (2000), 273 – 299.

[135] Schauenburg, P. - The monoidal center construction and bimodules, J. Pure Applied Algebra 158 (2001), 325 – 346.

[136] Schauenburg, P. - The structure of Hopf algebras with a weak projection, Algebr. Repre- sent. Theory 3 (1999), 187 – 211.

[137] Silver, L. - Noncommutative localization and applications, J. Algebra 7 (1967), 44 – 76.

[138] Simson, D. - Coalgebras, comodules, pseudocompact algebras and tame comodule type, Colloq. Math. 90 (2001), 101 – 150.

[139] Sweedler, M.E. - Hopf Algebras, Benjamin New York, 1969

[140] Szlachanyi,´ K. - Finite quantum groupoids and inclusions of ﬁnite type, Fields Inst. Comm. 30 (2001), 393 – 407.

[141] Szep,´ J. - Uber die als Produkt zweier Untergruppen darstellbaren endlichen Gruppen. Comment. Math. Helv. 22 (1949), 31 – 33.

[142] Takeuchi, M. - Free Hopf algebras generated by coalgebras, J. Math. Soc. Japan 23 (1971), 561 – 582.

[143] Takeuchi, M. - Matched pairs of groups and bismash products of Hopf algebras, Comm. Algebra 9 (1981), 841 – 882.

[144] Takeuchi, M. - Representations of the Hopf algebra U(n), AMS Contemporary Math. 441 (2007), 155 – 174.

[145] Vaes, S., Vainerman, L. - Extensions of locally compact quantum groups and the bicrossed product construction, Adv. Math. 175 (2003), 1 – 101.

[146] Wiegold, J., Williamson, A. G. - The factorisation of the alternating and symmetric groups. Math. Z. 175 (1980), 171 – 179.

[147] Zhu, Y. - Hopf algebras of prime dimension, Int. Math. Res. Notes 1 (1994), 53 – 59.

197 Index

(., /)-cocycle, 156 canonical matched pair, 184 (p, f) - left skew pairing, 122 category of descent data, 12 (p, f) - right skew pairing, 121 category of left-left A-Hopf modules, 55 (u, v) - skew braiding, 122 category of left-left Yetter-Drinfel’d modules, A-coring, 12 56 A-form of a Hopf algebra, 136 coalgebra,2 A ⊗ Aop-Yetter-Drinfeld module, vii, 43 coalgebra lazy 1-cocycle, 83 r-deformation of a Hopf algebra, 157 coassociative,2 r-deformation of a group, 185 coboundary, 117 (co)complete category,4 cocycle condition, 77 (co)equalizer,5 cofree coalgebra,1 (co)limit,4 cohomologous crossed systems, 116 (co)locally small category,4 coinduction functor,7 (co)product,5 comultiplication,2 (co)reﬂective category,6 convolution,2 (co)well-powered category,4 coquasitriangular Hopf algebra, 99 corestriction functor,7 adjiont functors,6 counit,2 algebra,2 couniversal problem,7 antipode,2 crossed product of a Hopf algebra acting on an Azumaya algebra, 11 algebra, 68 crossed product of Hopf algebras, 113 bialgebra,2 crossed system of Hopf algebras, 113 bialgebraHopf algebra extending structure, 70 descent datum, 12 bicomodule,1 Drinfel’d double, 138 bicrossed product of Hopf algebras, 68 Borel subalgebra, 111 epimorphism,4 braided Hopf algebra, 99 equivalent (., /)-cocycles, 156 braiding, 10 extending datum of a bialgebra, 69

198 INDEX extending datum of a group, 61 smash product, 113 extending structures problem, viii, 56 strict monoidal category, 10 extension of finite descent type, 155 strict onoidal functor,9 subobject,4 factorization A-form, 154 subspace of coinvariants, 55 factorization A-form of a group, 184 Sweedler notation,1 factorization index, 155 Sweedler’s canonical coring, 12 factorization problem,x, 58, 129, 130 Sweedler’s Hopf algebra, 163 Faithfully Flat Descent Theorem, 12 symmetric cocycle, 118 first cohomology group of a coalgebra,1 symmetry, 10 first lazy cohomology group, 83 fundamental theorem for Hopf modules, 55 Taft algebra,3 tensor algebra,1 generalized (u, v) - skew braiding, 100 tensor category,8 generalized (p,f) - left skew pairing , 100 trivial action, 55 generalized (p,f) - right skew pairing, 99 trivial cocycle, 113 generalized quantum double, 108 trivial coextension of a coalgebra, 28 generalized quaternion algebra, 40 twisted module condition, 77 group algebra,3 unified product of A and Ω(A), 69 Hopf algebra,2 unified product of groups, 62 universal enveloping algebra,3 Lie algebra,3 universal problem,6 locally presentable category,4 universal solution of the co-universal problem, 7 monoidal category,8 universal solution of the universal problem,6 monoidal equivalence, 10 monoidal functor,9 weak right center, 10 monomorphism,4 natural monoidal isomorphism, 10 natural monoidal transformation,9 normal coalgebra map, 88 normal Hopf subalgebra, 114 prebraiding, 10

Radford biproduct, 56 right A-module coalgebra, 55 right center, 10 right Yetter-Drinfeld Ae-module, 44 rigid extension, 155, 185 separable functor, 11 skew pairing, 99 small category,4

199