Journal of Algebra 219, 466᎐505Ž. 1999 Article ID jabr.1999.7896, available online at http:rrwww.idealibrary.com on
Cross Product Bialgebras, I
Yuri Bespalov
Bogolyubo¨ Institute for Theoretical Physics, Metrologichna Str. 14-b, View metadata, citation and similar papers at core.ac.uk brought to you by CORE Kie¨ 252143, Ukraine E-mail: [email protected] provided by Elsevier - Publisher Connector
and
Bernhard Drabant
Faculty of Mathematics, DAMTP, Uni¨ersity of Cambridge, Sil¨er Street, Cambridge CB3 9EW, United Kingdom E-mail: [email protected]
Communicated by Susan Montgomery
Received March 26, 1998
The subject of this article is bialgebra factorizations or cross product bialgebras without cocycles. We establish a theory characterizing cross product bialgebras universally in terms of projections and injections. Especially all known types of biproduct, double cross product, and bicross product bialgebras can be described by this theory. Furthermore the theory provides new families ofŽ. cocycle-free cross product bialgebras. Besides the universal characterization we find an equivalent Ž.co modular description of certain types of cross product bialgebras in terms of so-called Hopf data. With the help of Hopf data construction we recover again all known cross product bialgebras as well as new and more general types of cross product bialgebras. We are working in the general setting of braided monoidal categories, which allows us to apply our results in particular to the braided category of Hopf bimodules over a Hopf algebra. Majid’s double biproduct is seen to be a twisting of a certain tensor product bialgebra in this category. This resembles the case of the Drinfel’d double which can be constructed as a twist of a specific cross product. ᮊ 1999 Academic Press
INTRODUCTION
In recent years variousŽ. cocycle-free cross products with bialgebra structure have been investigated by several authorswx 34, 28, 22 . The
466 0021-8693r99 $30.00 Copyright ᮊ 1999 by Academic Press All rights of reproduction in any form reserved. CROSS PRODUCT BIALGEBRAS 467 different types such as tensor product bialgebra, biproduct, double cross product, and bicross product bialgebra are characterized each by a univer- sal formulation in terms of specific projections and injection of the particular tensorands into the cross product. The tensorands show interre- latedŽ. co module structures which are compatible with these universal properties and which allow a reconstruction of the cross product. The cross products are therefore equivalently characterized by either of the two descriptions. The multiplication and the comultiplication of the different cross products have a form similar to the multiplication and the comultipli- cation of the tensor product bialgebra except that the tensor transposition is replaced by a more complicated morphism with particular properties. TheŽ. co unit is given by the canonical tensor product Ž. co unit. Up to these common aspects the defining relations of the several types of cross products seem to be different. The question arises if there exists at all a possibility to describe the different cross products as different versions of a single unifying theory which equivalently characterizes cross product bial- gebras universally and in aŽ. co modular manner. The present article is concerned with this question and will give an affirmative answer. Based on the above-mentioned common properties of cross products we define cross product bialgebras or bialgebra admissible tuplesŽ. BAT . We show that there are equivalent descriptions of cross product bialgebras either by certain idempotents or by coalgebra projections and algebra injections obeying specific relations. However, it is not clear if a necessary and sufficient formulation by some interrelatedŽ. co module structures of the particular tensor factors exists as well. For these purposes we restrict the consideration to BATs where both the algebra and coalgebra structure of the tensorands is respected at least by one of the four projections or injections such that it is at the same time an algebra and a coalgebra morphism. The corresponding cross product bialgebras will be called trivalent cross product bialgebras. This is a sufficiently general class of cross product bialgebras to cover all the known cross products ofw 34, 28, 22x . Trivalent cross product bialgebras admit a universal characterization as well. On the other hand we define so-called Hopf data. A Hopf datum is a couple of objects which are both algebras and coalgebras and which are mutualŽ. co modules obeying certain compatibility relations. One can show that a certain Hopf datum structure is canonically inherited on any BAT. Conversely Hopf data induce an algebra and a coalgebra on the tensor s m product B B12B which strongly resembles the definition of a cross product bialgebra. However, there is a priori no compatibility of both structures rendering the Hopf datum a bialgebra. However, Hopf data obey the fundamental recursive identities f s ⌽Ž.f of Proposition 2.9 for s ⌬ ( s m ( m ⌿ m ( ⌬ m ⌬ both f BBm and f Ž.Ž.Ž.m BBBBm id , BBBBid . This fact leads us to the definition of so-called recursive Hopf data. 468 BESPALOV AND DRABANT
Recursive Hopf data turn out to be bialgebras. We also define recursive Hopf data with finite order and show that a special class of recursive Hopf dataŽ. with order F 2 , called trivalent Hopf data, are in one-to-one correspondence with trivalent cross product bialgebras. Hence the classifi- cation of cross product bialgebras either byŽ. co modular or by universal properties according towx 28 has been achieved for all trivalent cross product bialgebras in terms of trivalent Hopf data. The new classification scheme covers all the known types of cross products with bialgebra structurewx 34, 28, 22 . And for the most general trivalent Hopf data it provides a new family of cross product bialgebras which had not yet been studied in the literature so far.1 At the end of Section 2 we will apply our results in particular to Radford’s four-parameter Hopf algebra Hn, q, N, introduced inwx 29 . It turns out that it is a biproduct bialgebra over the sub-Hopf group algebra kCNnof H , q, N, . Since we are working throughout in very general types of braided categories, we can apply our results to the special case of the braided cat- egory of Hopf bimodules over a Hopf algebraŽ possibly in a braided category, too. . We demonstrate that Majid’s double biproductwx 26 is a bialgebra twist of a certain tensor product bialgebra in the category of Hopf bimodules. An example of double biproduct bialgebra is Lusztig’s construction of the quantum group U wx17 . A more thorough investigation of Hopf data and cross product bialge- bras in Hopf bimodule categories will be presented in a forthcoming work. Another application of our results shows that the braided matched pair formulation in terms of a certain pairing only works if the mutual braiding of the two objects of the pair is involutive. This confirms in some sense a similar observation inwx 26 . In Section 1 we give a survey of previous results, notation, and conven- tions which we need in the following. In particular we recall outcomes of wx4, 5 . We use graphical calculus for braided categories. The subsections on Hopf bimodules and twisting will be needed only in Section 3. Section 2 is devoted to the main subject of the article. We define bialgebra admissible tuplesŽ. BAT or cross product bialgebras, trivalent cross product bialge- bras, andŽ. recursive Hopf data Ž of finite order . . It turns out that Hopf data with a trivialŽ. co action are recursive and of order F 2. They will be called trivalent Hopf data. We show that trivalent Hopf data and trivalent cross product bialgebras are equivalent. They generalize the known ‘‘classi- cal’’ cross products which will be recovered as certain special examples. The results of Section 2 will be applied in Section 3. Using results ofwx 4 we demonstrate that the double biproductwx 26 can be obtained as a bialgebra
1A special variant of a trivalent Hopf datum has been studied inwx 2 which uniformly describes biproducts and bicross products in the symmetric category of vector spaces. CROSS PRODUCT BIALGEBRAS 469 twist from the tensor product of two Hopf bimodule bialgebras. We show that the braided version of the matched pairingwx 25 yields a matched pair if and only if the braiding of the two tensorands is involutive. A matched pair is a special kind of Hopf datum studied in Section 2.
1. PRELIMINARIES
We presume the reader’s knowledge of the theory of braided monoidal categories. Braided categories have been introduced in the work of Joyal and Streetwx 14, 15 . Since then they were studied intensively by many authors. For an introduction to the theory of braided categories we recommend having a short look at the above-mentioned articles or stan- dard references on quantum groups and braided categorieswx 10, 35, 16, 23 . Because of Mac Lane’s coherence theorem for monoidal categoriesw 20, 21x we may restrict our consideration to strict braided categories. In our article we denote categories by caligraphic letters C, D, etc. For a braided m monoidal category C the tensor product is denoted by C , the unit object | C ⌿ by C , and the braiding by . If it is clear from the context we omit the index C at the various symbols. Henceforth we consider braided categories which admit split idempotentswx 4, 18, 19 ; for each idempotent ⌸ s ⌸ 2 : M ª M of any object M in C there exists an object M⌸ and a pair of Ž. ( s ( s ⌸ morphisms i⌸⌸ , p such that p⌸⌸i idM ⌸ and i ⌸⌸p . This is not a severe restriction of the categories under consideration since every braided category can be canonically embedded into a braided category which admits split idempotentswx 4, 18 . We use and investigate algebraic structures in braided categories. We suppose that the reader is familiar with the generalization of algebraic structures to braided categories. Essentially we are working with algebras, coalgebras, bialgebras, Hopf algebras, modules, comodules, bimodules, and bicomodules in braided categorieswx 18, 23, 25 . Structures like Hopf bimod- ules or crossed modules will be reviewed in the following. We use through- out the symbol m for the multiplication and for the unit of an algebra, ⌬ for the comultiplication, and ⑀ for the counit of a coalgebra, S for the antipode of a Hopf algebra, for theŽ. left or right action of an algebra on a module, and for theŽ. left or right coaction of a coalgebra on a m ª | comodule. We call a morphism : M N C in C a pairing of M and N. The graphical calculus forŽ. strict braided monoidal categoriesw 14, 13, 16, 30, 31, 23, 35x will be used throughout the paper. We compose morphisms from up to down, i.e., the domains of the morphisms are at the top and the codomains are at the bottom of the graphics. Tensor products are represented horizontally in the corresponding order. We present our own conventionswx 4, 5 in Fig. 1 and omit an assignment to a specific object 470 BESPALOV AND DRABANT
FIG. 1. Graphical presentation of multiplication m, unit , comultiplication ⌬, counit ⑀, ⌿ antipode S, left action lr, right action , left coaction l, right coaction r, braiding , inverse braiding ⌿y1 , pairing , and convolution inverse y Ž.if it exists . if there is no fear of confusion. To elucidate graphical calculus we will represent below the bialgebra axiom of multiplicativity of the comultiplica- tion both in the ordinary way of composition and by graphical symbols. ⌬(m s Ž.Ž.Ž.m m m ( id m ⌿ m id ( ⌬ m ⌬
The results of the following subsections on Hopf bimodules and twisting will be needed in Section 3. They are not relevant for the central part of the article presented in Section 2.
Hopf Bimodules Hopf bimodules over a bialgebra B in C are B-bimodules and B-bicom- odules such that the actions are bicomodule morphisms through the diagonal coactions on tensor products of comodules and the canonical comodule structure on B wx4. B-Hopf bimodules and bimodule᎐bicomod- B B ule morphisms constitute the category BBC . For the symmetric category of k-vector spaces Hopf bimodules have been introduced inwx 36 under the name bicovariant bimodules. Suppose that H Is a Hopf algebra in C. Then there exists a tensor H H wx bifunctor rendering HHC aŽ. braided monoidal category 4, 5a, b . The proper formulation of the corresponding theorem requires two auxiliary ᭪ I H H H H bifunctors and on HHC . Two objects X and Y of HHC yield the ˙ I s m H-Hopf bimodule X ˙ Y X Y with diagonal left and right actions X I˙ YX I˙ Y X I˙ Y s d, ldand , r , and with induced left and right coactions i, l XXm I˙ YXs m ᭪ lYid and i, rXrid . The Hopf bimodule X Y is obtained CROSS PRODUCT BIALGEBRAS 471 by categorical dualization of the previous structures. For Hopf bimodule ᭪ s I s m morphisms f and g we define f g f ˙ g f g. Then the categories ŽH C H ᭪.ŽH C H I. HHC , and HHC , ˙ are semi-monoidal, i.e., they are categories which are almost monoidal, except that the unit object and the relations involving H H it are not required. For the definition of the braiding of HHC we use ⌰ ᭪ ªⅷ I op ⌰ [ Ž.YXm ( the natural transformation : ˙ given by X, Ylr m ⌿ m ( XYm m ª m Ž.id HX, YHlrid Ž.: X Y Y X. In the following theo- H H wx rem the braided monoidal structure of HHC is described 4, 5a, b . H H ¨ THEOREM 1.1. The category HHC of Hopf bimodules o er H is monoidal. m The unit object is the canonical Hopf bimodule H, and the tensor product H is uniquely definedŽ. up to monoidal equi¨alence by one of the following equi¨alent conditions for any pair of H-Hopf bimodules X and Y.
ⅷ m ¨ The H-Hopf bimodule X H Y is the tensor product o er H of the H ᭪ ª m ( underlying modules, and the canonical morphism X, YH: X Y X Y m H HH ᭪ ªⅷ m X HHY is functorial in CHH, i.e., : . ⅷ m ¨ The H-Hopf bimodule X H Y is the cotensor product o er H of the H I ( m underlying comodules, and the canonical morphism X, YH: X Y X HY ª I H C HH I ªⅷ I X ˙˙Y is functorial in HHC , i.e., : H . The corresponding natural morphisms H and H obey the identity
H ( H s X m Y ( m ⌿ m ( X m Y X , YX, YrlXHHYrlŽ.Ž.Žid id Ž..1.1.
H H H H H CH ⌿ The category HHC is pre-braided through the pre-braiding X, Y uniquely defined by the condition
H H H (H CH ⌿ ( H s ⌰ Y , XX, YX, YX, Y . It is braided if the antipode of H is an isomorphism in C. Another concept closely related to Hopf bimodules are crossed modules wx36, 37, 3, 4 . The connection of both notions had been studied in wx 36 and was reformulated inwx 32 for symmetric categories of modules over commu- tative rings. The general investigation for braided categories which admit split idempotents can be found inwx 4 . A right crossed module over the Hopf algebra H is an object M in C which is both right H-module and right H-comodule such that the follow- ing identity holds: 472 BESPALOV AND DRABANT
IdentityŽ. 1.2 is the graphical representation of the equation
m ( m ⌿ m ( m ⌬ Ž.ŽrHm id MM, HHrHid .Ž. s m ( ⌿ m ( m ( Ž.Ž.Ž.id MHm H, MHid id Hrr ( ⌿ m ( m ⌬ Ž.Ž.M , HHid id MH.
The right H-crossed modules and the corresponding module᎐comodule H morphisms form a category which is denoted by DYŽ.C H . In this notation D stands for Drinfel’d, who introduced the quantum double DHŽ.of a Hopf algebra H, and Y stands for Yetter, who identified the category of representations of DHŽ.with the category of H-crossed modules. There- fore crossed modules are sometimes called Drinfel’d᎐Yetter modules or ᎐ H Yetter Drinfel’d modules. DYŽ.C H is a monoidal category through the tensor product and the unit object of C, and the diagonalŽ. co actions for tensor products of crossed moduleswx 3, 4 . In the following we will outline the relation of Hopf bimodules and crossed moduleswx 4, 5 . If H is a Hopf algebra in the category C and Ž.X, rlrl, , , is an H-Hopf bimodule ( | m ( |I then there exists an object HHHX such that X X, HHX X, ( s | m ( ª ( | m andXX p i id HX , whereX p: X X HHXX X and i: |I ( ª ( | m HHX X X X are the corresponding universal morphisms. y H H ª H [ The assignment HHHŽ.: C DYŽ.C H, which is given through HŽ.X ( ( m m ( ( ŽŽ.HXX,p rXi id HX ,Ž.. p id H rXi for an object X, and through s ( ( ª HYXŽ.f p f i for a Hopf bimodule morphism f: X Y, defines a functor into the category of crossed modules. Conversely a full inclusion h y H ªH H functor H Ž.: DY Ž.C HHHC of the category of right H-crossed modules into the category of H-Hopf bimodules is defined by H h Ž.X s m HmXH mXH mXH mX Ž H X, i, li, , ld, , rd, , r . for any right crossed module X and h s m by H Ž.f id H f for any crossed module morphism f. The action HmX HmX i, ldis the left action induced by H and , r is the diagonal action of HmX the tensor product H-module. In the dual way the coactions i, l and HmX wx d, r are defined. The following theorem holds 4 .
THEOREM 1.2. Let H be a Hopf algebra in C with isomorphic antipode. H H H ¨ Then the categories DYŽ.C HHHand C are braided monoidal equi alent by
HhŽ.y 6 DY C H H C H . Ž.H 6 HH H Žy.
If not otherwise mentioned we subsequently assume that the antipode of the Hopf algebra H is an isomorphism in C. Remark 1. A mirror symmetric result corresponding to Theorem 1.2 holds for left H-crossed modules and H-Hopf bimodules. Henceforth we CROSS PRODUCT BIALGEBRAS 473
( ( will denote the idempotentsXX i p and i Xp X of a Hopf bimodule X by ⌸ ⌸ ⌸ s X ( m ( X XXand , respectively. Explicitly it holds that XlHXlŽ.S id ⌸ s X ( m ( X wx and XrŽ.id XHrS 4. This observation leads to the following useful lemma.
LEMMA 1.3. Suppose that X and Y are H-Hopf bimodules and f, g: ᭪ ª I X Y X ˙ Y are Hopf bimodule morphisms. Then the identity H H H CH ⌿ ( H ( ⌸ m ⌸ s H (⌿ ( ⌸ m ⌸ X , YX, YXŽ. Y Y, XX, YX Ž.Ž. Y 1.3 holds. The identity ⌸ m ⌸ ( ( ⌸ m ⌸ s ⌸ m ⌸ ( ( ⌸ m ⌸ Ž.Ž.Ž.Ž.Ž.XYXYXYXYf g 1.4 implies f s g. Proof. The composition of both sides ofŽ. 1.3 with the monomorphism ⌰ ( ⌸ m ⌸ s XYm ( Y, XYobviously leads to the identity , XXŽ.Ž Y r l. m ⌿ m ( XYm ( ⌸ m ⌸ Ž.id XHHYrid Ž lX.Ž Y ., which in turn coincides with ⌿ ( ⌸ m ⌸ Y, XXŽ. Y. To prove the second statement of the lemma observe that any Hopf bimodule morphism f: X᭪Y ª X IY can be expressed X s ( ⌸ m ⌸ ˙ Y s in terms of f f Ž.XYand subsequently in terms of f ⌸ m ⌸ ( ( ⌸ m ⌸ Ž.Ž.XYXYf as
XY wx where ad, r , ad, l are theŽ. braided adjoint coactions 3, 4 . The second identity ofŽ. 1.5 is derived from the module properties of f. In a similar way one obtains the third identity ofŽ. 1.5 . RelationŽ. 1.3 is the braided counterpart of Woronowicz’s definition of braiding of Hopf bimodulesŽ seewx 36. . Finally we recall the first part of the canonical transformation procedure H H wx of bialgebras in HHC into bialgebras in C 4 which we need in the following. 474 BESPALOV AND DRABANT
PROPOSITION 1.4. Let H be a Hopf algebra in C. A bialgebra B s ⌬ ⑀ H H s Ž.B,mBB , , BB, in HHC can be turned into a bialgebra B ⌬ ⑀ ¨ Ž.B,mBB , , BB, in C, where the structure morphisms are gi en by s ( H s ( ⌬ s H (⌬ ⑀ s ⑀ (⑀ m BBBm , BBBHBB, , , BB, B HB. Ž.1.6
s ⌬ ⑀ H H s If B Ž.B,mBB , , BBB, , S is a Hopf algebra in HHC then B ⌬ ⑀ ¨ Ž.B,mBB , , BBB, , S is a Hopf algebra in C with antipode SB gi en s ( s ( s ( m ( m by SBBBS S r HBS r HBS , where S Br HlHrHŽ.Žid S m ( m ( id BHS .Žid Hrl . .
Twisting In this subsection we present the twisting construction for bialgebras in a braided category C. We proceed along the lines ofwx 12, 24 . ª | Let C be a coalgebra and let : C C be a morphism into the unit object. Henceforth we will use the notation . f [ Ž. m f (⌬, f. [ Ž.f m (⌬ Ž.1.7 for any morphism f: C ª B in C. m ª | DEFINITION 1.5. If B is a bialgebra in C and : B B C is a morphism obeying the identities ( m s ( m Ž.Ž.id AA.m .m id , Ž.1.8 ( m s ⑀ s ( m Ž.id AA Ž.id Ž.1.9 then is called a 2-cocycle of the bialgebra B.If is a convolution invertible 2-cocycle, then the twist in m BBof the multiplication m is [ y defined by m BB.m . .If B is a Hopf algebra then the twist SBof the [ y s ( m (⌬ antipode S is given by SBBu.S .u , where u Ž.id BBBS . Remark 2. Under the condition of Definition 1.5 the first identity in Ž.1.9 holds if and only if the second one is valid. In analogy towx 12, 24 the following proposition can be verified in the braided case because nowhere in the proof the involutivity ⌿ 2 s id is needed. Therefore we will only sketch how to prove the proposition.
DEFINITION AND PROPOSITION 1.6. Let B be bialgebraŽ. Hopf algebra m ª | ¨ [ and let : B B C be an in ertible 2-cocycle. Then B ⌬ ⑀ Ž B,mBB , , BB, , ŽS B.. with the twisted multiplication m BŽand twisted antipode SB .Ž. is again a bialgebra Hopf algebra . B is called the twisted bialgebraŽ. Hopf algebra of B obtained by the twist . CROSS PRODUCT BIALGEBRAS 475
Proof. At first we will demonstrate that the bialgebra axiom L [ ⌬(m s Ž.Ž.Ž.m m m ( id m ⌿ m id ( ⌬ m ⌬ [ R for B is equivalent to the bial- gebra axion L s Rfor B . This follows from the identities Ž.L. s .L and Ž.R . s .R. Second, using the previous fact we show that the associativity Al [ m(Ž.Ž.m m id s m( id m m \ Ar of B is equivalent to l r the associativity of B , denoted by As A . This is proved with the help l ( m s ( m l of the identities Ž A .Ž. Žid AA.m .. Ž Ž.m id .. . A and rr ( m s ( m Ž.ŽŽA . .m id AA ..ŽŽid .m .. . A which result fromŽ. 1.8 .
2. CROSS PRODUCT BIALGEBRAS AND HOPF DATA
Section 2 is the central part of the article. We define cross product bialgebras or bialgebra admissible tuplesŽ. BAT and Hopf data. We consider certain specializations of these definitions, which we call trivalent cross product bialgebras and recursive Hopf data, respectively. Trivalent cross product bialgebras form a sufficiently general class to cover the cross product bialgebras ofwx 34, 28, 22 . Additionally there arise new explicit examples of trivalent cross product bialgebras. All of them will be com- pletely classified in terms of recursive Hopf data. Therefore an equivalent description either through interrelatedŽ. co module structures or through universal projector decompositions is found.
Cross Product Bialgebras
Suppose now there are two objects B12and B in C, and morphisms m ª m m ª m 1, 2: B 1B 2B 2B 1and 2, 1: B 2B 1B 1B 2 . ⌬ ⑀ ⌬ ⑀ DEFINITION 2.1. We call ŽŽ B11111,m , , , .Ž, B 2,m 2222 , , , ., 1, 2, 2, 1.Ža bialgebra admissible tuple BAT. in the category C if ⌬ ⑀ g Ž.Bjjj,m , is an algebra and Ž.Bjjj, , is a coalgebra for j Ä41, 2 , such that ⑀ ( s id| , and the object B m B is a bialgebra through jj C 12 m s m m m ( id m m id , s m , x12Ž.Ž.B122,1 BX12 Ž.2.1 ⌬ s id m m id ( ⌬ m ⌬ , ⑀ s ⑀ m ⑀ . x Ž.B121,2 B Ž.12 x12
This bialgebra will be called the cross product bialgebra associated with the bialgebra admissible tuple Ž.B121,22,1, B , , and is denoted by B = B . 1 1, 2 2, 1 2 One observes that the definition of a cross product bialgebra differs from the usual definition of a canonical tensor product bialgebraŽ in a symmetric category. only through the substitution of the tensor transposi- tion by the morphisms 1, 2and 2, 1. 476 BESPALOV AND DRABANT
Of course the known cross products with bialgebra structure are cross product bialgebras in the sense of Definition 2.1 if the structures of the objects B12and B and of the morphisms 1,22and ,1are chosen correctly. The following proposition allows us to express cross product bialgebras in terms of idempotents or projections and injections.
PROPOSITION 2.2. Let A be a bialgebra in C, then the following statements are equi¨alent.
1. A is a bialgebra isomorphic to a cross product bialgebra B = B . 1 1, 2 2, 1 2 ⌸ ⌸ g 2. There are idempotents 12, EndŽ.A such that ( ⌸ m ⌸ s ⌸ ( ( ⌸ m ⌸ ⌸ ( s m AjŽ. j jAjjm Ž., jAA, ⌸ m ⌸ (⌬ s ⌸ m ⌸ (⌬ (⌸ ⑀ (⌸ s ⑀ Ž.Ž.jjA jjAjAjA,
for j g Ä41, 2 , and the sequence
( ⌸ m⌸⌸m⌸ ( ⌬
m Ž.Ž.6 6 A m AAAA 12 12A m A ⌸ m ⌸ m is a splitting of the idempotent 12of A A.
3. There exist objects B12 and B in C which are at the same time algebras and coalgebras, and algebra morphisms i,jjcoalgebra morphisms p, ip ªjjª ( s g BjjA B , such that p jjBi id for j Ä41, 2 , and the morphisms ( m m ª mj (⌬ ª m ¨ m A Ž.i121i:B B 2A and Ž.p 1p 2 A: A B12B are in erse to each other. « ⌸ g Proof. Ž.2 Ž.3 . Since j for j Ä41, 2 are idempotents there are ª ª ⌸ morphisms ijj : B A and pj : A B jwhich split j. We define [ ( ( m m jjAjjp m Ž.i i, [ ( jjAp , ⌬ [ m (⌬ ( jjjAjŽ.p p i, ⑀ [ ⑀ ( jAji
g for j Ä41, 2 . One immediately verifies that Ž.Bjjj,m , are algebras and ⌬ ⑀ Ž.Bjjj, , are coalgebras, and that ij are algebra morphisms and pj are g ⌸ coalgebra morphisms for j Ä41, 2 . Because j are idempotents it follows ( m ( m (⌬ s ( ⌸ m ⌸ ( ⌸ m ⌸ (⌬ s that m A Ž.Ž.i12i p 1p 2AAm Ž.Ž.1212A m ( idA , where the last equation follows by assumption. SimilarlyŽ. p12p ⌬ (m (Ž.i m i s idm is proven. AA12 B12B CROSS PRODUCT BIALGEBRAS 477
« g ⌸ s ( Ž.3 Ž.2 . For j Ä41, 2 we consider the idempotents jjji p . State- mentŽ. 2 is then proven easily with the help of the assumed properties of ijj and p . Ž.1 « Ž.3 . Let : B = B ª A be the isomorphism of bialgebras. 1 1, 2 2, 1 2 Then in particular
( m id s id m 2,1Ž. 2 BB112 Ž.2.2 ( id m s m id 2,1 Ž.B2211B
⑀ ⑀ and dually analogous for 1, 2and 1, 2 . We define the morphisms
i [ ( id m ,p[ id m ⑀ (y1 , 1 Ž.B1121 Ž.B 2 Ž.2.3 i [ ( m id , p [ ⑀ m id (y1 . 21Ž.B2221 Ž.B
UsingŽ. 2.3 one verifies without problems that ij are algebra morphisms. In g a dual manner it is proven that pj are coalgebra morphisms for j Ä41, 2 . Since ⑀ ( s id| s ⑀ ( it follows that p (i s id , j g Ä41, 2 . Be- 11 22 jj Bj cause is a bialgebra isomorphisms it holds that
m ( i m i s m ( m ( id m m m id A Ž.12 AB Ž.Ž.1221 B s ( ( m m m s m BBB = Ž.id 21id B , 1 1,2 2,1 21 2 whereŽ. 2.2 has been used in the third equation. Dually one obtains m (⌬ s y1 Ž.p12p A . « [ ( m Ž.3 Ž.1 . By assumption the isomorphisms m A Ž.i12i induces a [ m bialgebra structure on B B12B through
s y1 ( ( m ⌬ s y1 m y1 (⌬ ( m BAm Ž., BŽ.A, Ž.2.4 s y1 ( ⑀ s ⑀ ( BA, BA.
We have to show that B with the structureŽ. 2.4 is a cross product bialgebra. At first we prove that theŽ. co units of B are given by the tensor s m products of the particularŽ. co units of B12and B . It holds B Žp1 (⌬ ( s ( m ( p2 . AAp1 A p2 A because A is a bialgebra. Furthermore s ( s ( A i11i 22is satisfied since i 1 and i 2 are algebra morphisms. s m ( s Combining these two equations then yields B 12since pjji id B g ⑀ s ⑀ m ⑀ j for j Ä41, 2 by assumption. Dually B 12can be proven. Now we are going to prove that B has the structure of a cross product bialgebra if we use the morphisms s Ž.⑀ m id m id m ⑀ (⌬ and s 1, 2 1 BB212 B 2, 1 478 BESPALOV AND DRABANT
m (Ž. m id m id m . Thereto we need the auxiliary relations B 1 BB212
id m m ( y1 m p s y1 (m ( id m ⌸ , Ž.B1 22Ž. AAŽ.2 Ž.2.5 m m id ( p m y1 s y1 (m ( ⌸ m id , Ž.1 B1 Ž.1 A Ž.1 A which can be proven easily because is a bialgebra isomorphism and i1 and i2 are algebra morphisms. UsingŽ. 2.5 we show that mB is indeed a multiplication of the formŽ. 2.1 ,
m m m ( id m m id Ž.12Ž.B122,1 B s m m id ( p m y1 ( id m m ( i m i m Ž.1 B2 Ž.1 Ž.Ž.AA12 s y1 ( ( m m A Ž. s m.B Ž. 2.6
In the first equation ofŽ.Ž. 2.6 , 2.5 has been used two times. In the second equality we applied p (i s id and againŽ. 2.5 . The third equation of 11 B1 Ž.2.6 holds by definition. Similarly it can be shown by dualization that the comultiplication is that of a cross product bialgebra, i.e., ⌬ s Žid m BB1 m id .Ž( ⌬ m ⌬ .. Hence ŽB , B , , .is a BAT and B s 1, 2 B2 12 121,22,1 B = B its corresponding cross product bialgebra. 1 1, 2 2, 1 2 ⑀ ⑀ Since identitiesŽ. 2.2 and their dual analogues for 1, 2and 1, 2 hold for cross product bialgebras, we immediately derive
COROLLARY 2.3. Let B = B be a cross product bialgebra. Then 1 1, 2 2, 1 2 m id and id m are algebra morphisms, and ⑀ m id , id m ⑀ 1 BB2121BB212 are coalgebra morphisms. It is not clear if the very general definition of a cross product bialgebra is in one-to-one correspondence with a description in terms of pairs of Ž.co algebras with certain interrelated compatibleŽ. co module structures. Hence a classification of cross product bialgebras in the sense ofwx 28 may not succeed at this general level. But for reasons of classification and reconstruction this aspect is important. The known cross products with bialgebra structurewx 34, 28, 22 admit such a description. For later use we will therefore define trivalent cross product bialgebras as follows.
DEFINITION 2.4. A cross product bialgebra B = B is called 1 1, 2 2, 1 2 m m ⑀ m trivalent if at least one of the morphisms 1 idBB , id 21, idB , m ⑀ 21 2 id B 2 is both an algebra and a coalgebra morphism. In a slight abuse 1 3 of notation we denote the corresponding bialgebra by B = B with- 1 1, 2 2, 1 2 out indication of the specific algebra᎐coalgebra morphism. CROSS PRODUCT BIALGEBRAS 479
In particular all cross products inwx 34, 28, 22 are trivalent. Up to now we investigated universality of cross product bialgebras. In the following subsection we study cross product bialgebras from aŽ. co modular point of view.
Hopf Data
DEFINITION 2.5. A Hopf datum Ž.B12, B , ll, , rr, in C consists of two objects B12and B which are both counital algebras and unital coalgebras, and Ž.B1, l is a left B21-module, Ž.B , l is a left B2-comodule, Ž.B2 , r is a right B11-module, and Ž.B , r is a right B1-comodule obeying the identities
( m s (⑀ s m ⑀ ( ⑀ ( s ⑀ m ⑀ s ⑀ ( r Ž.21id 2121 Ž.id l , 2 r 211l , ( m s (⑀ s ⑀ m ( ( s m s ( l Ž.id21 122 Ž.id 1rr, 221l 1 ,
algebra᎐coalgebra compatibility,
module᎐comodule compatibility,
module᎐algebra compatibility, 480 BESPALOV AND DRABANT
comodule᎐coalgebra compatibility,
module᎐coalgebra compatibility,
comodule᎐algebra compatibility. At first sight the defining relations of Hopf data seem to be rather complicated and impenetrable. However, all the compatibility identities only relate the differentŽ. co algebra and Ž. co module structures. Besides there are two remarkable symmetries of the definition of Hopf data. The first one is the usual categorical duality in conjunction with the transfor- l ⌬ l ⑀ l l mation ‘‘m ,’’ ‘‘ ,’’ ‘‘ ll,’’ and ‘‘ rr.’’ The second one is a kind of mirror symmetry with respect to a vertical axis of the defining equations considered as graphics in three-dimensional space, followed by the transformation of the indices ‘‘1 l 2’’ and ‘‘l l r.’’ This observation considerably simplifies subsequent considerations and calculations. In a first step we recover canonicalŽ. co algebra structures of Hopf datum. PROPOSITION 2.6. LetŽ. B12, B , ll, , rr, be a Hopf datum. We define
s m Then B B12B is both an algebra and a coalgebra through the structure morphisms
m s m m m ( id m m id , s m , B Ž.12Ž.B122,1 BB12 Ž.2.8 ⌬ s id m m id ( ⌬ m ⌬ , ⑀ s ⑀ m ⑀ . BBŽ.121,2 B Ž.12 B 12 CROSS PRODUCT BIALGEBRAS 481
Proof. It is a well-known fact that the necessary and sufficient condi- tions for Ž.B,mBB , being an algebra are given by the equations
( m m id s id m m ( m id ( id m 2,1Ž.Ž.Ž.Ž. 2 BB1122,1BB222,1 ( id m m s m m id ( id m ( m id 2,1 Ž.Ž.Ž.Ž.B2211BB12,1 2,1 B 1 Ž.2.9 ( m id s id m 2,1Ž. 2 BB112 ( id m s m id . 2,1 Ž.B2211B
The verification of the third and fourth equations ofŽ. 2.9 can be done straightforwardly usingŽ. 2.7 and the defining relations of a Hopf datum. The second equation ofŽ. 2.9 will be proven graphically,
The first identity ofŽ. 2.10 uses the algebra᎐coalgebra compatibility of Definition 2.5. In the second equation the module᎐algebra compatibility is
used. The third equality holds because B12is a left B -module, and the fourth identity is true because of the module᎐coalgebra compatibility. This proves the second equation ofŽ. 2.9 . The first identity of Ž. 2.9 can be verified similarly. Hence Ž.B,mBB , is an algebra. It will be proven dually ⌬ ⑀ that Ž.B, BB, is a coalgebra. Under the conditions of Proposition 2.6 one proves that m id , 1 B2 id m are algebra morphisms, and ⑀ m id , id m ⑀ are coalgebra B1 21BB212 morphisms. The result of Proposition 2.6 strongly resembles the definition of a BAT. However, there is a priori no compatibility of the algebra and s m the coalgebra structure of B B12B rendering B a cross product bialgebra. On the other hand the following proposition is easily proved. 482 BESPALOV AND DRABANT
PROPOSITION 2.7. A BATŽ. B121,22,1, B , , yields a Hopf datum Ž.B12, B , ll, , rr, through
s id m ⑀ ( , s ⑀ m id ( lBŽ.1222,1 r Ž.1 B 2,1 Ž.2.11 s ( id m , s ( m id . l 1,2 Ž.B122 r 1,2 Ž. 1 B
Con¨ersely according to Eqs.Ž. 2.7 in Proposition 2.6 the resulting Hopf datum = can be transformed into the cross product bialgebra B1 B2 because it s s 1, 2 2, 1 holds that 1,2 1,2and 2,1 2,1. Proof. Equations of the formŽ.Ž 2.9 and their dual form. hold in particular for 2, 1 Ž.and 1, 2 . Hence the Ž. co module properties, the module᎐algebra and the comodule᎐coalgebra compatibility of Definition 2.5 can be derived for theŽ. co actions in Ž 2.11 . . The unital coalgebra
structure and the counital algebra structure of B12and B , as well as the relations in Definition 2.5 involving theŽ. co actions and theŽ. co units can be shown easily. By assumption B = B is a bialgebra and therefore 1 1, 2 2, 1 2 it holds
⌬==(m s m =m m =( id m m ⌿ m m m id m ( ⌬==m ⌬ . Ž.Ž.B12BB 1212B , B BB 12B Ž. Ž.2.12
Then one deduces the algebra᎐coalgebra compatibility of a Hopf datum by either composingŽ. 2.12 with ( Žid m m id m .and Ž id m ⑀ m B112 B 2 B 12 id m ⑀ .Ž( or with ( m id m m id .Ž and ⑀ m id m ⑀ m B1221B 1 B21 B21 id .Ž(. The module᎐comodule compatibility is derived from 2.12. by B2 composition with (Ž. m id m id m and Ž.⑀ m id m id m ⑀ (. 1 BB2121BB222 If one applies (Ž. m id m m id and Ž.⑀ m id m id m ⑀ ( to 1 B221 B 1 BB 212 Ž.2.12 one gets the first identity of the comodule᎐algebra compatibility. The second equation of the comodule᎐algebra compatibility is derived by ( m m m ⑀ m m m composingŽ. 2.12 with Žid B 2 id B 21.Žand id BBid ⑀ ( ᎐ 11 21 2 . . The module coalgebra compatibility is proven dually. The composi- ( m m m ⑀ m m m ⑀ ( tion ofŽ. 2.12 with Žid B 21idB .Ž and 1 id BBid 2 . s 122 s 1 yields 1, 2 1, 2 . The identity 2, 1 2, 1 is derived fromŽ. 2.12 with (Ž. m id m id m andŽ. id m ⑀ m ⑀ m id (. This concludes the 1 BB212 B121 B2 proof of the proposition. Hence Hopf data are more general objects than BATs, and by Proposi- tion 2.7 we may interpret cross product bialgebras as some subclass of Hopf data. Subsequently we will show that two noteworthy recursive identities are satisfied for Hopf data. Although these identities are rather involved, they possess the above-mentioned dual and mirror symmetries. To avoid complications we represent the identities graphically. CROSS PRODUCT BIALGEBRAS 483
DEFINITION 2.8. Let Ž.B12, B , ll, , rr, be a Hopf datum in C and g m m m let f End C Ž.B1212B B B be any endomorphism in C. Then the ⌽ m m m ª m m m mapping : End C Ž.Ž.B1212B B B End C B1212B B B is given by
In the following proposition the fundamental recursive relation for Hopf data will be derived. PROPOSITION 2.9. LetŽ. B12, B , ll, , rr, be a Hopf datum in C and [ ⌬ ( [ m ( m consider the endomorphisms f1 BBm and f2 Ž.Žm BBBm id ⌿ m ( ⌬ m ⌬ ⌬ B, BBBBid .Ž ., where m Band Bare defined according to Ž.2.8 . s ⌽ s ⌽ Then it holds that f1122Ž.f and f Ž.f . s ⌬ ( Proof. For f1 BBm we obtain the result through theŽ. graphical identities 484 BESPALOV AND DRABANT
The first identity ofŽ. 2.14 uses the algebra᎐coalgebra compatibilities of Definition 2.5. In the second identity ofŽ. 2.14 we used the module᎐coal- gebra and the comodule᎐algebra compatibilities.Ž. Co Associativity is ap- plied to derive the third equation ofŽ. 2.14 . s m ( m ⌿ m ( ⌬ m ⌬ For f2 Ž.Ž.Ž.m BBBBm id , BBBBid the proof is given by the graphical equalities
where the first identity requires the algebra᎐coalgebra compatibility, the second uses the module᎐algebra and the comodule᎐coalgebra compatibili- ties, as well as theŽ. co module properties of B12and B . In the third equation we applied associativity and again theŽ. co module properties of
B12and B .
Remark 3. Observe that we did not need the complete list of defining relations of a Hopf datum for the deduction of Proposition 2.9. We only needed that B12and B are both algebras and coalgebras, B1is a left y ᎐ B221-Ž. co module, B is a right B Ž.co module, the algebra coalgebra compatibility, the module᎐coalgebra compatibility, the comodule᎐algebra compatibility, the module᎐algebra compatibility, and the comodule᎐coal- gebra compatibility. In particular we did not use the module᎐comodule compatibility.
From Propositions 2.6 and 2.1 we conclude that Hopf data are more general objects than BATs and therefore do not correspond to them directly. In the following we will restrict our considerations to so-called recursive Hopf data. They do not necessarily imply universal characteriza- tion. For that reason a more special kind of recursive Hopf data, so-called trivalent Hopf data, will be introduced below. We will verify that trivalent Hopf data and trivalent cross product bialgebras are indeed equivalent CROSS PRODUCT BIALGEBRAS 485
notions which provide aŽ. co modular and universal description of cross product bialgebras. The resulting theory will turn out to be general enough to unify all the ‘‘classical’’ cross products ofwx 34, 28, 22 . And it will even generate a new family of cross product bialgebras which can be described in this manner. The equivalent formulation of cross product bialgebras either by interrelatedŽ. co module structures or by certain universal projec- tions and injections therefore will be provided by our theory. DEFINITION 2.10. Let Ž.B12, B , ll, , rr, be a Hopf datum in C and [ (⑀ m m (⑀ define the idempotent 11id B mB 22. Suppose that for m m m21 every endomorphism f of B1212B B B there exists a non-negative g ގ ⌽ n s ⌽ n ( ( integer n 0 such that Ž.f Ž.f . Then the Hopf datum ⌳ [ g ގ N ⌽ n s Ž.B12, B , ll, , rr, is called recursive. If Än 0 Ž.f ⌽ n ( ( ᭙ g m m m Ž.f f End C ŽB1212B B B .4 is a non-empty set such [ g ⌳ that n01minÄ4n exists, we call Ž.B , B2, ll, , rr, a recursive Hopf datum of order n0. A consequence of the structure of ⌽ is the following lemma. ¨ LEMMA 2.11. For e ery Hopf datumŽ. B12, B , ll, , rr, it holds that (⌽ ( s ( ( g m m m Ž.f f for any f End C ŽB1212B B B .. If ¨ ⌽ m s ⌽ n ᭙ G Ž.B12, B , ll, , rr, is recursi e of order n then Ž.f Ž.f m n. Proof. The first statement is verified straightforwardly using Hopf datum properties and the structure of ⌽ according to Definition 2.8. Then it follows for a Hopf datum of order n that ⌽ nŽ.f s ⌽ nŽ. ( f ( s ⌽ nnnŽŽ..ŽŽ..Ž. (⌽ f ( s ⌽⌽f s ⌽ q1 f . A Hopf datum Ž.B12, B , ll, , rr, is non-trivial if B12or B is not | isomorphic to C . ¨ LEMMA 2.12. A recursi e Hopf datumŽ. B12, B , ll, , rr, of finite order is non-tri¨ial if and only if its order is greater than 0. Proof. If the order is 0 then f s ( f ( and in particular for s s (⑀ ( | f idB mB mB mBBiii one shows that id . Therefore B C for s 1212 i | s i Ä41, 2 . Conversely if B12and B are isomorphic to C then id m m m and one concludes that the order of the Hopf datum is 0. B1212B B B Remark 4. We will henceforth assume that Hopf data are non-trivial so that the order is greater than 0 if it exists. For special categories recursivity of a Hopf datum implies its finite order.
PROPOSITION 2.13. Suppose that C is a category of modules o¨er a ¨ commutati e ring , andŽ. B12, B , ll, , rr, is a Hopf datum in C with B12 and B free -modules of finite rank. ThenŽ. B12, B , ll, , rr, is recursi¨e if and only if it is recursi¨e of finite order. 486 BESPALOV AND DRABANT
¨ Proof. Suppose that Ž.B12, B , ll, , rr, is recursive and let Ä4 iig I be m m m g aŽ. finite basis of B1212B B B . Then every endomorphism f m m m s и ␦ ¨ End C Ž.B1212B B B can be written as f ÝI, Iif , jiˆ, ji, where fŽ. s и ¨ ␦ ¨ s ␦ и ¨ [ g ގ N ⌽ n ␦ s ÝIif , jjand ˆ i, jkŽ. i, kj. Define n00minÄn Ž.ˆi, j ⌽ n (␦ ( ᭙ g << Ž.ˆi, j i, j I4, which exists since I is finite and the Hopf datum is supposed to be recursive. Then one immediately concludes that
the order of the Hopf datum is n0. The most striking aspect of recursive Hopf data is the bialgebra struc- m ture of the corresponding tensor productŽ. co algebra B12B . ¨ THEOREM 2.14. IfŽ. B12, B , ll, , rr, is a recursi e Hopf datum s m ⌬ then B B12B equipped with the structure morphisms m,BB, B and ⑀ according to Proposition 2.6 is a bialgebra. It will be denoted by B s B l r B j B . 1 lr2 ⌬ Proof. Because of Proposition 2.6 we only have to prove that B is an algebra morphism. With the help of Proposition 2.9 and the recursivity of g ގ s ⌽ n s the Hopf datum we derive for some n the identities f11Ž.f ⌽ n ( ( s ⌽ n s ⌽ n ( ( s ⌬ ( Ž.f1222and f Ž.f Ž.f , where f 1BBm and s m ( m ⌿ m ( ⌬ m ⌬ ( ( f2 Ž.Ž.Ž.m BBBBm id , BBBBid . The relation f1 s ( ( ᎐ f2 holds by the module comodule compatibility of the Hopf s data. Therefore f12f , which proves the theorem. In the next proposition we will introduce trivalent Hopf data. DEFINITION AND PROPOSITION 2.15. Suppose thatŽ. B12, B , ll, , rr, is a Hopf datum in C. Then one of theŽ. co actions of Ž B12, B , ll, , rr, . ¨ m m ⑀ m is tri ial if and only if one of the morphisms 1 idBB , id 21, idB , m ⑀ 21 ¨ 2 or id B 2 is both an algebra and a coalgebra morphism. If these equi alent 1 ¨ ¨ conditions hold, we callŽ. B12, B , ll, , rr, a tri alent Hopf datum. Tri a- lent Hopf data are recursi¨e of order F 2. Therefore the corresponding 3 lr lr bialgebra B j B exists and will be denoted by B j B . 1 lr21lr2 s Proof. Suppose that lBis trivial. Then fromŽ. 2.8 we conclude m m ( m m ( ⌿ m ( m ⌬ m Žm 12m .Žid BBrB Žid .Ž , BBBid .Žid 1 .idB . . m11⑀ ( s ( 2m112⑀ m m ⑀ m2 ThereforeŽ. id B 2 m B m1 Žid B 2 id B 2 .Žand id B ⑀ ( s 11m ⑀ 11 2 .ŽB 1, which shows that id B 2 .is an algebra morphism. Be- m ⑀ 1 cause of Proposition 2.6,Ž. id B 2 is also a coalgebra morphism. Con- m ⑀ 1 versely ifŽ. id B 2 is an algebra morphism it holds in particular that m ⑀ ( 1 ( m m m s ( m ⑀ m m Ž.Žid B 2 m B 1 id BBid 21.Žm id B 2 id B ⑀ (12 m m m 11 s ⑀ m 1 21.Ž id BBid 2 .from which the triviality l 2 idBl of 21 1 is derived. Since l is trivial the identity
⌽ f s ⌽ id m m m (⑀ ( f ( (⑀ m idm m 2.15 Ž. Ž.Ž.B121B B 22Ž. 11 B212B B Ž. CROSS PRODUCT BIALGEBRAS 487
can be verified directly for any endomorphism f. From the general structure of ⌽ as given in Definition 2.8 we derive
id m m m (⑀ (⌽ f ( (⑀ m id m m Ž.Ž.B121B B 22Ž. 11 B212B B
s id m m m (⑀ Ž.B121B B 22
(⌽ (⑀ m id m m ( f ( id m m m (⑀ Ž.Ž.11 B212B BBŽ.121B B 22
( (⑀ m idm m . 2.16 Ž.11 B212B B Ž.
UsingŽ. 2.15 two times, Ž. 2.16 , and then againŽ. 2.15 one obtains the result ⌽ 2 s ⌽ 2 ( ( g m m m Ž.f Ž.f for any f End C ŽB1212B B B .. Hence the order of the Hopf datum is F 2. Because of the dual and mirror symme- tries of Hopf data the proof of the proposition follows analogously for any otherŽ. co action being trivial.
Remark 5. One verifies easily thatŽ. 2.15 can be obtained under the following conditions, which are weaker than the assumption of triviality of one of theŽ. co actions,
or similar conditions involving lror . Then it follows quite analogously as in Proposition 2.15 that the Hopf datum is recursive of order F 2. Using the notion of Proposition 2.6 these conditions can be reformulated as conditions of 1, 2 and 2, 1. These are conditions of a BAT since the Hopf datum is recursive. Therefore a generalization of Theorem 2.16 below can be obtained in this wayŽ see also the notion ‘‘strong Hopf datum’’ inwx 6. .
The following theorem demonstrates that trivalent Hopf data and triva- lent cross product bialgebras coincide. This shows that a unified theory of cross product bialgebras has been found which provides universal and Ž.co modular characterization equivalently. 488 BESPALOV AND DRABANT
THEOREM 2.16. Suppose that A is a bialgebra in C. Then the following statements are equi¨alent. ¨ 1. There is a tri alent Hopf datumŽ. B12, B , ll, , rr, such that the 3 lr corresponding bialgebra B j B is a bialgebra isomorphic to A. 1 lr2 2. A is bialgebra isomorphic to a tri¨alent cross product bialgebra 3 B = B . 1 1, 2 2, 1 2 ª 3. There are algebra morphisms i:jjB A and coalgebra morph- isms p: A ª B such that p (i s id for j g Ä41, 2 , m (Ž.i m i s jj jjBj A12 m (⌬ y1 ŽŽp12p .A . , and one of the morphisms i,i,p,p12 1 2is both an algebra and a coalgebra morphism. ⌸ ⌸ g 4. There are idempotents 12, EndŽ.A such that
( ⌸ m ⌸ s ⌸ ( ( ⌸ m ⌸ ⌸ ( s m AjŽ. j jAjjm Ž., jAA, ⌸ m ⌸ (⌬ s ⌸ m ⌸ (⌬ (⌸ ⑀ (⌸ s ⑀ Ž.Ž.jjA jjAjAjA,
for e¨ery j g Ä41, 2 , the sequence
( ⌸ m⌸⌸m⌸ ( ⌬
m Ž.Ž.6 6 A m AAAA 12 12A m A
⌸ m ⌸ m is a splitting of the idempotent 12of A A, and one of the idempo- ⌸ ⌸ tents 12, is either an algebra or a coalgebra morphism. Proof. Essentially the proof of the theorem has been done in Proposi- tions 2.2, 2.6, 2.7, and 2.15. We only have to show the additionalŽ. co alge- bra properties of the corresponding morphisms or the triviality of the correspondingŽ. co actions. Ž.3 « Ž.1 . From Proposition 2.2 it follows especially that A is isomor- phic to a cross product bialgebra B s B = B through the bialgebra 1 1, 2 2, 1 2 isomorphism : B = B ª A, s m (Ž.i m i . Then there is a 1 1, 2 2, 1 2 A 12 l r Hopf datum such that B s B j B s B = B because of Propo- 1 lr211, 2 2, 1 2 sition 2.7. Suppose that p1 is an algebra and a coalgebra morphism. Then ( ( s ( m ( m ( s ( ( m p1 m B m11Ž.Ž.p p 11i i 2m B p1 m A Ž.. Therefore ( ( m ( ( s ( m ( ( m ( m ( m m11112Ž.Ž.p i p i m B m11m m 1111211Žp i p i p i p (i.Ž . Since p (i s id and p (i s id m ⑀ .(y1 ( (Ž. m id 12 11 B1112 B 21B2 s (⑀ the identity s Ž.id m ⑀ ( s ⑀ m id follows then from 12 lB1122B Ž.2.8 and Ž 2.11 . . « Ž.1 Ž.2 . If there is a trivalent Hopf datum with trivial left action l then s Ž.id m ⑀ ( s ⑀ m id and thereforeŽ. id m ⑀ (m s lB1122BB 12 B m (Ž.id m ⑀ m id m ⑀ andŽ id m ⑀ .( s , which implies that 1 B112 B 2 B 12 B 1 Ž.id m ⑀ is an algebra morphism. Because of Proposition 2.6 it is also a B1 2 coalgebra morphism. CROSS PRODUCT BIALGEBRAS 489
Ž.2 « Ž.4 . Suppose that Ž id m ⑀ . is an algebra morphism. Using B1 2 Proposition 2.2 and the bialgebra isomorphism : B = B ª A we 1 1, 2 2, 1 2 obtain ⌸ s i (p s (Ž.Ž.id m ( id m ⑀ (y1 and therefore ⌸ 111 B112 B 21 is an algebra morphism. « ⌸ ⌸ ( s ( ( s Ž.4 Ž.3.If 11is an algebra morphism then m1i1p1m A ( ( m ( s ( ( m ⌸ ( s ( ( s s m A Ž.Ž.i11p i 11p i 1m 1p 1p 1 and 1A i11p AA ( i11because i1 is an algebra morphism. Since i1 is monomorphic one concludes that p1 is an algebra morphism and also a coalgebra morphism by Proposition 2.2. Thus the theorem is proven for a particular case. Because of dual and mirror symmetry all other cases can be verified analogously. Theorem 2.16 shows that there exists a one-to-one correspondence of trivalent cross product bialgebras and trivalent Hopf data. In addition both notations are equivalent to a description in terms of a certain projector decomposition. Since Definition 2.4 of a trivalent cross product bialgebra is a generalization of the cross products with bialgebra structure according towx 34, 28, 22 , we can express all of them in a unified manner through trivalent Hopf data. Moreover the most general trivalent Hopf data give rise to a new family ofŽ. trivalent cross product bialgebras. For a better understanding of the theory we list five special examples of trivalent Hopf data in the sequel which cover all the other cases because of the dual and mirror symmetries. The discussion of the different types of trivalent Hopf data Ž.B12, B , ll, , rr, will be taken up with the help lr
of the table , where the particular entries take values 0 or 1 depen- lr dent on whether theŽ. co action is trivial or not. Thus by definition maximally three entries in the table take the value 1.
00 COROLLARY 2.17. . All the actions and coactions are tri¨ial. 00 Then the corresponding Hopf datum is equi¨alently gi¨en by the following data. B and B are bialgebras in C, ⌿ (⌿ s idm , and 12 B21, BB 12, BB 1B 2 3 lr B j B is the canonical tensor product bialgebra B m B . 1 lr212 10 . The tri¨alent Hopf datum is gi¨en through the following data. 10
B21 is a bialgebra in C and B is a B2-crossed comodule bialgebra in 3 B2 wx lr l DYŽ.C 3, 4 . Then B j B s B j B is the braided ¨ersion of the B2 1 lr21 l 2 crossed product or biproduct wx28, 4 .
00 . The tri¨alent Hopf datumŽ. B , B , , is a braided ¨ersion 11 12lr wx of the matched pair 22 . Explicitly, B12 and B are bialgebras in C, B1 is a left 490 BESPALOV AND DRABANT
B22-module coalgebra, B is a right B1-module coalgebra, and the following defining relations are fulfilled.
The corresponding bialgebra B j B is a braided ¨ersion of the double 1 lr2 cross product wx22 . The bialgebra structure reads
10 . The tri¨alent Hopf datum is gi¨en byŽ. B , B , , , where 01 12lr
B12 and B are bialgebras in C, B1 is a left B 2-comodule coalgebra, Bisa2 right B1-module algebra, and the following defining relations are fulfilled.
⑀ ( s ⑀ m ⑀ ( s m 2 r 21, r 221, CROSS PRODUCT BIALGEBRAS 491
l The affiliated bialgebra B j B is a braided ¨ersion of the bicross product 1 r 2 wx22 . The bialgebra structure is gi¨en by
11 . This is the most general tri¨alent Hopf datumŽ B , B , 01 12 lrr, , .. B122 is a bialgebra in C and a left B -comodule. B is a counital right B1-module-comodule algebra and a unital coalgebra, and the following defining identities hold. ⑀ ( s ⑀ m ⑀ ( s m 2 r 21, l 121 ⑀ m id ( s (⑀ ,idm ⑀ ( s (⑀ Ž.2 Br1212 Ž.B 1 l 21
l r The structure of the resulting bialgebra B j Bisgi¨en by 1 r 2 492 BESPALOV AND DRABANT
Proof. Straightforward evaluations of Definition 2.5 using the particu- lar trivialŽ. co actions. In the following we will discuss two examples which are closely related to Hopf algebras constructed by Ore extensionswx 1 . The first example is an infinite Hopf algebra whereas the second example is Radford’s finite-di- mensional four-parameter Hopf algebrawx 29 . Both Hopf algebra types turn 01 out to be trivalent cross product bialgebras of type and therefore 01 are biproduct Hopf algebras according towx 28 . Closely related are the examples studied in the forthcoming paperwx 9 , where the universal aspects have been considered.
EXAMPLE 1. Suppose C is an abelian group and k is an algebraically U closed field with charŽ.k s 0. Let C be the character group of C and let g ގ s g t U s UU t . Assume further that g Ž.g1,..., gt C and g Ž g1 ,..., gt . g U t U ŽC . , where at least one of the gijand one of the g is non-trivial, and [ U и s g glrgg lŽ. r with glrg rl 1 for any l, r Ä41,...,t . Let the algebra U t HCŽ ,t, g, g . be generated by the group C and let the generators Ä4xiis1 be subject to the additional relations и s U и и s и x jjc g Ž.cc x jand x jkjkkjx gx x Ž.2.18 for any j, k g Ä41,...,t and c g C. Then the following relations define a U Hopf algebra structure on HCŽ , t, g, g ..
⌬Ž.c s c m c, ⑀ Ž.c s 1, Sc Ž.s cy1 , Ž.2.19 ⌬ s m q | m ⑀ s sy и y1 Ž.x jjjx g x j, Ž.x j0, Sx Ž.jx jjg
U for all j, k g Ä41,...,t and c g C. Every element of HCŽ , t, g, g . is a s и finite sum of the form h Ýcg Cjcc fxŽ., where fx Ž. jcis a Ž non-com- mutative. polynomial in Ä4x j . The Hopf algebra is finite-dimensional if C is sy g finite and g jj 1 for all j Ä41,...,t . [ Now we define B1 kC the group Hopf algebra of C with comultipli- ⌬ ⑀ [ r и cation 112and counit on C. Let B be the algebra B 2kx²Ä4jjk:Žx x s и ᭙ g gxjk kx j j, k Ä41,...,t .. Then the following definitions yield k-linear morphisms.
U U B ¨ HC, t, g, g B ¨ HC, t, g, g i: 12Ž.i: Ž. 12½½c ¬ cx¬ x U ª U ª Ž.2.20 HCŽ., t, g, g B1 HCŽ., t, g, g B2 p: p: 12c и fx ¬ ⑀ fx c c и fx ¬ ⑀ cfx ½ Ž.jjŽ. Ž. ½ Ž.jj Ž.Ž. CROSS PRODUCT BIALGEBRAS 493
Straightforward calculations show that i12 is a Hopf algebra morphism, i is an algebra morphism, and p1 is an algebra morphism. Then one concludes ⌬ m ⌬ ( (⌬ easily that p11 is a coalgebra morphism since Ž.1p1 and p1are algebra morphism, and the equality of both morphisms has to be proven wx ⌬ on the generators only. According to 9 we define the comultiplication 2 ⑀ ⌬ [ m (⌬( ⑀ [ ⑀ ( and counit 222222of B as Ž.p p i and 2i 2 . Using U again the fact that every element of HCŽ , t, g, g . is a finite sum of the s и form h Ýcg Cjcc fxŽ., one finds
⌬ (p h s ⑀ c ⌬ fx 22Ž.Ý Ž.2Ž. Žj .c cgC s ⌬ fx Ý Ž.Ž.j c cgC s p m p (⌬ fx Ý Ž.Ž.22Ž.j c cgC and on the other hand
p m p (⌬ h s p m p c m c и⌬ fx Ž.Ž.Ž.Ž.Ž.22 Ý 22ž/Ž.j c cgC s p m p ⌬ fx . Ý Ž.Ž.22ž/Ž.j c cgC ⌬ ( s m (⌬ ⌬ Hence 22p Ž.p 2p 2 . Then it follows that 2is coassociative and p22 is a coalgebra morphism. Furthermore p is not an algebra morphism. и s и s ⑀ Suppose the converse, then p2Žc x j .p22 Ž.c p Žx jj . Ž.cx. On the и s U и s U ⑀ ⑀ s other hand p2Ž.c x j p2 ŽgcxjjŽ.c . gc jŽ.Ž.cx j. Therefore Ž.cx j U ⑀ U gcjjŽ.Ž.cx. But by assumption there exists a non-trivial g j which then leads to a contradiction. Similarly, by the existence of a non-trivial g j it can be shown that i2 is not a coalgebra morphism. m (⌬ s ( m y1 m Now we prove thatŽ.ŽŽ.. p12p m i12i . Observe thatŽ p1 (⌬( s | m ( id.Ž.i 22id i because the corresponding identity holds on the m (⌬( generators and then the statement follows sinceŽ. p12id i and | m ( m Ž.id i2 are algebra morphisms. Then for any finite sum Ýcg C c g m fxŽ.jc B12B we have
p m p (⌬(m( i m i c m fx Ž.12 Ž. 12Ý Ž.j c cgC s p m p (⌬ c и fx Ý Ž.12Ž. Ž.j c cgC s c m | и p m p (⌬ fx Ý Ž.Ž.Ž.12Ž.j c cgC 494 BESPALOV AND DRABANT
s c m | и | m p fx Ý Ž.ž/2 Ž. Ž.j c cgC s c m fx . Ý Ž.j c cgC
m (⌬ ( m ThusŽ. p12p is a left inverse of m Ž.i12i . Similarly the right invertibility can be proven. Therefore all conditions of Theorem 2.16 hold, and the morphisms 1, 2 and 2, 1 of the corresponding cross product bialgebra are given by s Ž.⑀ m id m id m ⑀ (⌬ and s 1, 2 1 BB212 B 2, 1 m (Ž. m id m id m Žsee the proof of Proposition 2.2. . Using B 1 BB212 Ž.2.11 yields the Ž non-trivial .Ž. co actions
m s и s m (⌬ r Ž.Ž.x c p2 x c and r Ž.Ž.Ž.x p21p x .
From Corollary 2.17 eventually follows U PROPOSITION 2.18. The Hopf algebra HŽ C, t, g, g. is isomorphic to the r biproduct bialgebra B j B . In particular B is a right B -crossed module 1 r 221 bialgebra. The following example is Radford’s four-parameter Hopf algebrawx 29 . U Its structure resembles that of the Hopf algebra HCŽ , t, g, g . discussed in Example 1.
EXAMPLE 2Ž. Radford’s Four-Parameter Hopf Algebra . Suppose again that k is an algebraically closed field with charŽ.k s 0. Let n, N, be positive integers such that n N N, and - n. Let q be a primitive nth root s r of unity and let q be an rth root of unity, where r n Ž.n, . Set CN to be the cyclic group of order N and g a generating element of CN . Then the four-parameter Hopf algebra Hn, q, N, is generated by the group algebra kCN and the generator x subject to the additional relations
x r s 0 and x и g s qg и x.Ž. 2.21
On the generators the comultiplication ⌬, counit ⑀, and antipode S are given by
⌬Ž.g s g m g, ⑀ Ž.g s 1, Sg Ž.s gy1 , ⌬Ž.x s x m | q gy m x, ⑀ Ž.x s 0, Sx Ž.syg и x,
Similarly as in the previous example one shows that every element of s ry1, Ny1 m и l Hn, q, N, can be represented in the form b Ýms0, ls0 m, l x g .In [ particular Hn, q, N, is finite-dimensional. Let B11be the algebra B CROSS PRODUCT BIALGEBRAS 495
r r [ kx²:Žx . and let B22be the group Hopf algebra B kCN . Then B1 becomes a coalgebra through
m ⌬ mls m m mylm⑀ s ␦ 11Ž.x Ý x x and Ž.x m ,0, Ž 2.22 . ž/l q ls0 where the q-binomial is
m Ž.m p! [ , Ž.s pppp![ Ž.1 и Ž.2 и иии и Ž.s , ž/l p Ž.l pp! Žm y l .! 1 y p s Ž.0 pp[ 1, and Ž.s [ . 1 y p
Using the properties of the q-binomials it is evident thatŽ. 2.22 renders B1 a coalgebra. Then it can be proven that the following definitions yield k-linear homomorphisms i12 , i , p 1 , and p 2 .
¨ B ¨ H B1 Hn, q, N , 2 n, q, N , i:12i: ½ x mm¬ x ½ g ll¬ g Ž.2.23 ª U ª Hn, q, N , B1 HCŽ., t, g, g B2 p: p: 12mlи ¬ m mlи ¬ ␦ l ½ x g x ½ x g m ,0g
Obviously i12 , i , and p 2 are algebra morphisms since they preserve the relations of the algebras Hn, q, N, , B12and B . Furthermore i22 and p are coalgebra homomorphisms since the corresponding identities hold for ⌬ ( the generators g and x. Finally, p11 is a coalgebra morphism because mlи s ⌬ m p11Ž x g .Žx . and
m (⌬ mlи s m ⌬ m и lm l Ž.p11p Ž.x g Ž.Ž.p11p Ž.x Ž.g g s m m | q y m m и llm Ž.Žp11p Ž.x g x .Ž.g g m s m m jlи y Ž myj. m myjlи Ý Ž.p11p Ž.x g x g s ž/j q j 0 m m y s Ý x jmm x j s ž/j q j 0 s ⌬ m 1Ž.x , 496 BESPALOV AND DRABANT
q mms m jmи yj where we used the q-binomial identity Ž.a b Ý js0 Ž.j a b if и s y1 и a b b a. This proves that p1 is a coalgebra morphism. Since by assumption q / 1 and g /| one concludes similarly as in Example 1
that i11 is no coalgebra morphism and p is no algebra morphism. Be- cause Hn, q, N, is finite-dimensional the subsequent relations prove that m (⌬ s ( m y1 Ž.ŽŽ..p12p m i12i, m (⌬( ( m m m l Ž.p12p m Ž.i 12i Ž.x g s m (⌬ m и l Ž.p12p Ž.x g s m ⌬ m и llm Ž.Ž.p12p Ž.x Ž.g g m s m m jlи y Ž myj. m myjlи Ý Ž.p12p Ž.x g x g s ž/j q j 0 m m y s Ý Ž.x jmm ⑀ Ž.xgjl s ž/j q j 0 s x m m g l. Hence again all conditions of Theorem 2.16 hold, and the morphisms 1, 2 and 2, 1 of the corresponding cross product bialgebra are given by s Ž.⑀ m id m id m ⑀ (⌬ and s m ( Ž m id m id m 1, 2 1 BB212 B 2, 1 B 1 BB21 mlm s ly mmm lmm s 21.Ž, or explicitly ,2x g . g x and 2,1Ž g x . qxyml mm g l. UsingŽ. 2.11 yields the Ž non-trivial .Ž. co actions
l m m s yml m m s y m m m llŽ.g x q x and Ž.x g x . We collect the previous results in the next proposition.
PROPOSITION 2.19. The four-parameter Hopf algebra Hn, q, N, is isomor- r j phic to the biproduct bialgebra B12 B . In particular B1 is a left B 2-cross module bialgebra. r
3. APPLICATIONS
In Section 3 we discuss further applications of the results of Section 2. The first example shows that Majid’s double biproductwx 26 essentially is a H H cross product bialgebra construction in the braided category HHC of Hopf bimodules over a Hopf algebra H. In the second subsection we show that the Drinfel’d double in a braided category can be reconstructed as a matched pair if and only if the braiding of the two tensor factors is involutive. This confirms in a certain sense the statements ofwx 26 . CROSS PRODUCT BIALGEBRAS 497
Double Biproduct Bialgebras We consider a Hopf algebra H. From Theorem 1.2 and its mirror symmetric version we know that every right H-crossed module B g H g H ObjŽŽ.DY C HH.Žand every left H-crossed module C Obj DYŽ..C yield s h s i H H Hopf bimodules X H B and Y C H in HHC , respectively. These special types of Hopf bimodules will be used later for the construction of H H the double biproduct as a twist of a tensor product bialgebra in HHC considered as a bialgebra in the category C according to Proposi- tion 1.4. g H g H LEMMA 3.1. Let B ObjŽŽ.DY C HH.Žand C Obj DYŽ..C be H- crossed modules. Then for the objects X s H h B and Y s C i H in the H H category HHC the identity
H H H H H CH H CH ⌿ ( ⌿ s id m X , YY, XXHY holds if and only if
⌿ (⌿ s B m C ( m ⌿ m ( B m C C , BB, CrlBHŽ.Ž.Žid , HCrid Ž. l.3.1. Proof. FromŽ. 1.3 and its dual version one deduces
H H H H ⌸ m ⌸ ( (H CH ⌿ (H CH ⌿ ( ( ⌸ m ⌸ Ž.XYX, YY, XX, YX, YXŽ. Y s ⌸ m ⌸ (⌿ ( H ( H (⌿ ( ⌸ m ⌸ Ž.XYY, XY, XY, XX, YXŽ.Ž. Y.3.2 ⌸ m ConditionŽ. 3.1 means that the right-hand side ofŽ. 3.2 equals to ŽX ⌸ ( ( ( ⌸ m ⌸ YX.Ž., YX, YX Y. Then we use Lemma 1.3 to derive
H H (H CH ⌿ 2 ( s ( X , YX, YX, YX, YX, Y . Since H is an epimorphism and H is a monomorphism the sufficient X, YX, Y H H H H H CH H CH part of the lemma is proved. Conversely if ⌿ ( ⌿ s id m X, YY, XXH Y then Eq.Ž. 1.1 proves Ž. 3.1 . Lemma 3.1 is a braided version of the identityŽ. 58 inwx 26 which was one of the compatibility conditions for the construction of the double biprod- uct bialgebra. We suppose henceforth that B and C are bialgebras in H H DYŽ.C HHand DYŽ.C , respectively. Then according to Theorem 1.2 s h s i H H wx the objects X H B and Y C H are bialgebras in HHC . From 4 Ž.and Proposition 1.4 we know that the multiplications mXY and m are s ( H s uniquely determined. In particular m H h BXXm , XXand m ( m m H h BHXŽ.id i , where m Hh B is the multiplication of the crossed wx product algebra in C Žsee Corollary 2.17 and 4. andX i is the morphism described below Theorem 1.1. Similarly the multiplication mY and the 498 BESPALOV AND DRABANT comultiplications ⌬ and ⌬ can be calculated. The condition H H H H XY H CH H CH ⌿ ( ⌿ s idm of Lemma 3.1 allows the construction of the X, YY, XYH X m H H canonical tensor product bialgebra Y HHHX in C according to Corollary 2.17. Explicitly
H H H CH m m s Ž.m m m ( id m ⌿ m id , Y HXYHXYHXž/, YH X Ž.3.3 H H H CH ⌬ m s id m ⌿ m id (Ž.⌬ m ⌬ . Y HXYHYž/, XH X YH X
LEMMA 3.2. Suppose that the conditions of Lemma 3.1 are fulfilled for the H H bialgebras X and Y in HHC . Then the Hopf bimodule tensor product bialgebra m m m Y H X is identified with the object C H B through the canonical morphisms gi¨en by H s m m H s m ⌬ m Y , XCHBid m id , Y, XCHBid id .Ž. 3.4 [ m s m m H H We denote by Z Y HHX C H B the bialgebra in CH. Then H s m H m H s m H m Z, ZCXid , YBid , Z, ZCXid , YBid .Ž. 3.5
H H Proof. Without problems one verifies that Y, XYand , X according toŽ. 3.4 fulfill the required properties of Theorem 1.1. Using the iden- m ( m m tification Y H X C H B induced fromŽ. 3.4 the proof of Ž. 3.5 follows. According to Proposition 1.4 one obtains a bialgebra Z s C m H m B s m in C from the H-Hopf bimodule bialgebra Z Y H X. The explicit structure of Z is presented in the subsequent proposition.
PROPOSITION 3.3. Multiplication and comultiplication of the bialgebra Z s C m H m B are gi¨en by
s m m ⑀ s ⑀ m ⑀ m ⑀ The unit and counit read as ZCHBad ZCHB, ¨ m respecti ely. There are canonical bialgebra monomorphisms idCmHB: i ª m h ª C H Z, CHidmBC : H B Z, and bialgebra epimorphisms id mH m ⑀ ª i ⑀ m ª h m m BC: Z C H, idHmBC : Z H B. Additionally id H CROSS PRODUCT BIALGEBRAS 499
m ª m ⑀ m ª idB : C B Z is an algebra monomorphism, and idCHid B : Z C m B is a coalgebra epimorphism. Proof. From the considerations before Lemma 3.2 we know that the s m multiplication of Z Y H X is given by
H H s m ( m H CH ⌿ m m ZYHXYHXŽ.m m ž/id , YHid X , and according to Proposition 1.4 the multiplication of Z reads as s ( H m ZZZm , Z
H H s m ( mmH CH ⌿ m ( H Ž.m YHm Xž/id YH X, YHid X Z, Z
H H s m m ( mH CH ⌿ ( H m Ž.m CBCXH m ž/id , YX, YBidŽ 3.7. where we usedŽ. 3.5 in the third equation. FromŽ. 1.3 and from the Hopf H H H CH ⌿ H bimodule property of X, YXand , Y Ž.see Theorem 1.1 it follows that
H H H CH ⌿ ( H X , YX, Y
H H sH CH ⌿ ( H ( X ( m ⌸ ( X X , YB, ClHXlŽŽ.Ž.id m Y ( ⌸ m ( Y Ž.rYHrŽ.id . s ( m H (⌿ ( ⌸ m ⌸ m ( X m Y M Ž.id HYŽ., XX, YXŽ. Yid HŽ l r. s H ( Y m X ( m ⌿ m Y , XlŽ. rŽ.Žid HX, YHid 3.8. m m s Y H X ( Y H X m where M rlŽ id H.Ž.Ž. . Inserting 3.8 into 3.7 yields the ⌬ final result for mZZ . Dually can be derived. If it is clear from the context we will henceforth denote the bialgebra Z in C by C m H m B.
PROPOSITION 3.4. Let B and C be as in Lemma 3.2. Suppose that m ª | : B C C is a morphism in C satisfying the identities ( B m s ( m C Ž.Ž.rBid id Bl, ( m s Ž2. ( ⌿y1 (⌬ m Ž.id BCm Ž. B, BBid CmC , Ž.3.9
H y ( m s Ž2. ( DY ŽC .H ⌿ m ⌿ 1 (⌬ Ž.m BCid ž/B, BC, CC H Ž2. s ( m m DY ŽC .H ⌿ s m B ( ⌿ m where Ž.id BCid and B, BBrBŽid .Ž , B ( m B H wx [ ⑀ m m id HBr.Žid .Žis the braiding in DY C.H4. Then ˆ Ci H ⑀ m m H h B is a 2-cocycle of the bialgebra C H B from Proposition 3.3. If is con¨olution in¨ertible then ˆˆis con¨olution in¨ertible with in¨erse ys ⑀ m ym ⑀ C i HHh B. 500 BESPALOV AND DRABANT
Proof. The convolution invertibility and the cocycle propertyŽ. 1.9 for ⌬ ˆ, mZZ , and can be proven easily. The calculation of the left- and the right-hand sides ofŽ. 1.8 respectively yields ( B m ( m ( ⌿ m ( ⌬ m Ž.rCid ŽŽ.Ž.Ž.id B B, BCid BCid m m ( m ⌿ m Ž.Ž.Ž.id HBHid , CCid ( m ( ⌿ m ( m B m ⌬ Ž.Ž.Ž.id BHm H, BHid Ž.id Hr C. Ž.3.10 and ( m ( B m m Ž.m BCid Ž. rid BCid ( m ( ⌸ (⌬ m m Ž.Ž.Žid BB, BBid C .id HmBmC , Ž. 3.11 where the first and the second defining properties of inŽ. 3.9 have been used. With the help of the third equation ofŽ. 3.9 one can show that Ž 3.10 . equalsŽ. 3.11 . Remark 6. The identitiesŽ. 3.9 are braided versions ofw 26, Eqs.Ž. 56x . The following corollary is a straightforward consequence of Propositions 1.6, 3.3, and 3.4.
COROLLARY 3.5. Suppose that the pairing in Proposition 3.4 is in¨ert- ˆ ible. Then according to Proposition 1.6 the multiplication m CmHmB of the m m ¨ twisted bialgebraŽ. C H Bisgiˆ en by
qy where the pairing is presented by K and its con¨olution in¨erse y by K . Proof. Because of Proposition 3.4 we can apply Definition 1.5 and
Proposition 1.6 to mZ in Proposition 3.3. A straightforward calculation then yields the result. CROSS PRODUCT BIALGEBRAS 501
We will discuss in the following an example in which unavoidably cross product bialgebras in certain braided categories emerge.
EXAMPLE 3. Corollary 3.5 is aŽ. braided generalization of Majid’s double biproduct constructionwx 26 . It has been shown in w 26, 33 x that the quantum enveloping algebra U in terms of Lusztig’s constructionwx 17 is a double biproduct bialgebra. Explicitly, let Ž.I, и be a Cartan datum, and let Ž²:.X, Y, и , и be a root datum of type Ž.I, и . Given the commutative ring k s ޑŽ.q , let f be the k-algebra generated by I, factorized by the annihila- tor radical of the unique pairing Žи, и .: kI ²:= kI ²:ª k given in Proposi- wx tion 1.2.3 in 17 . Furthermore let U0 be the group algebra of Y over k. wx ( m m Then 17 , U f U0 f are isomorphic Hopf algebras. The algebra f is both a left and right U0-crossed module bialgebra. The bialgebra structure of f is induced by the algebra structure of f and by primitivity of all ²: wxs [ ގ wx elements of kI. From 17 we know that f g ގw I x f is an I -graded ¨ ¬ algebra. The root datum provides embeddings x: I X, i xi and y: ¨ ¬ I Y, i yi which in turn canonically induce homomorphisms of abelian wx wx groups x: ގ I ª X, ¬ x and y: ގ I ª Y, ¬ y . Then the left y 2 [ ² y, x < <: U0-crossed module structure of f is given by y q and [ y и r и << m g g liŽ.Ý g Ii Ži i 2 i. y , where y Y and f is homo- s << s << g ގwx g geneous of degree Ýig I i i I , i.e., f < <. The right y 1 [ ² y, x < <: [ U0-crossed module structure of f is y q and rŽ. y и r и <<m ' m m Ýig IiŽi i 2 i. y . Then U f U0 f is a double biproduct s s s bialgebra according to Corollary 3.5 with H U0 , C f, and B f, and the pairing s Ž.и, и . Although in the present example the base category C s k-mod is symmetric, the categories of H-crossed modules and H-Hopf bimodules are braided and the cross product bialgebra construction is within these categories. This emphasizes once again that the double biproductŽ even in ordinary symmetric categories. is a cross product bialgebra construction in a braided category.
Quantum Double Construction In Corollary 2.17 we discussed braided versions of matched pairs leading to braided double cross products. Fromwx 25 we know that two dually paired bialgebras in a symmetric category yield a matched pair from which a generalization of the Drinfel’d double can be reconstructed. Such a procedure had been discussed for braided categories inwx 26 . It was announced that a similar construction fails there since the braiding twists up and cannot be disentangled. The subsequent proposition confirms this observation in a certain sense. 502 BESPALOV AND DRABANT
PROPOSITION 3.6. Suppose that A and H are bialgebras in C which are и и m ª | paired by the pairing ²:, : H A C subject to the defining identities
и и ( m s и и ( m и и m ²:, Ž.m H id ²:, Žid ²:, id . ( m m ⌿(⌬ Ž.id id A и и ( m s и и ( m и и m ²:, Ž.id m A ²:, Žid ²:, id . Ž.3.13 ( ⌬ m m Ž.H id id и и ( m s ⑀ ²:, Ž.HAid и и ( m s ⑀ ²:, Ž.id AH.
Then the following statements hold.
1. If A and H are Hopf algebras and SA is an isomorphism in C then и и ¨ ¨ и и s и и ( m s ²:, is con olution in ertible. Explicitly ²:²:Ž, , SH id . и и ( m y1 ²:Ž, id SA .. 2. If ²:и , и is con¨olution in¨ertible we define
1s и и ym m и и ( m ⌿ m ( ⌬Ž2. m ⌬ Ž.²:, id ²:, Ž.id HmH , AHAid Ž. Ž.3.14 2s и и ym m и и ( m ⌿ m ( ⌬ m ⌬Ž2. Ž.²:, id ²:, Ž.id H , Am AHAid Ž., where ⌬Ž2. s Ž.⌬ m id (⌬. Then Ž. A, 2 is a left H-module andŽ. H, 1 is a right A-module. The tupleŽ. A, H, 1 , 2 is a matched pair as in Corollary ⌿ (⌿ s 2.17 if and only if H, AA, H id. Proof. Statement 1 is proved analogously as in the standard symmetric case. Without problems one verifies that 1 and 2 in statement 2 define ⌿ (⌿ s algebra actions. Now suppose that H, AA, HAidmH . It is not difficult to show that Ž.A, H, 1 , 2 is a matched pair because nearly everything works as in the classical symmetric casewx 22, 25 . Conversely if Ž A, H, 1 , 2.Ž.defined by 3.14 is a matched pair then the following identity has to be fulfilled because of the last equation inŽ. 2.17 . CROSS PRODUCT BIALGEBRAS 503
⌿ (⌿ s и и FromŽ. 3.15 we obtain A, HH, AHidm A by multiplying ²:, to the y left and ²:и , и to the right ofŽ. 3.15 using the product given byŽ. 1.7 .
4. CONCLUSIONS AND OUTLOOK
We defined Hopf data and cross product bialgebras very generally. Cross product bialgebras are Hopf data. A special class of Hopf data are recursive Hopf data probably with finite order. Recursive Hopf data are cross product bialgebras. We further restricted to trivalent Hopf data and trivalent cross product bialgebra. We showed the equivalence of both notions and provided a description of trivalent cross product bialgebras either throughŽ. co modular properties or by universal systems of certain projections and injections, respectively. Therefore the classification of trivalent cross product bialgebras in terms of trivalent Hopf data has been achieved. The known cross products with bialgebra structure fit into this new classification scheme. In addition new types of trivalent cross product bialgebras have been found which generalize all other types. However, explicit examples have not been found yet for these general types of trivalent cross product bialgebras. We have been working throughout in a braided monoidal setting which allowed us to apply the machinery of Hopf data and cross product bialgebras to braided categories. In particular we showed that the double biproduct bialgebras come from a certain tensor product bialgebra in the braided category of Hopf bimodules over a given Hopf algebra. A more general study of recursive Hopf data in Hopf bimodule categories will be published elsewhere. The structure ofŽ. recursive Hopf data shows to be symmetric under duality and reflection at a vertical axisᎏif one considers the defining identities as graphics in three-dimensional space. These symmetries will be somehow destroyed when considering trivalent Hopf data and trivalent cross product bialgebras and one might ask if such a breaking of symmetry is a generic feature of the theory of cross product bialgebras. Therefore it remains an open problem if the present setting is the most general one to describe cross product bialgebras byŽ. co modular properties or by universal systems of projection and injection morphisms equivalently. One could think of certain types of recursive Hopf dataŽ. with finite order or some other specializations of Hopf data which preserve the above-mentioned symmetries, to be good candidates for a more general framework. A possible generalization has been presented in Remark 5 although the symmetries will be destroyed in this case, too. 504 BESPALOV AND DRABANT
In the present article we studied cross product bialgebras without cocycles and cyclesŽ. or dual cocycles . In a forthcoming paperwx 6 we will apply similar techniques, and results fromwx 7, 8, 11, 27 to describe certain types of cross product bialgebras with cocycles in a cocyclicŽ. co modular way. Analogous statements as in Proposition 2.2 and Theorem 2.16 will be derived for rather general cross product bialgebras with cocycles. But the cocyclicŽ. co modular scheme of classification turns out to be much more subtle than in the present case. There might be different ways of restrict- ing the general setup of a cocycle cross product bialgebra to achieve different sorts of classification schemes.
REFERENCES
1. M. Beattie, S. Dascalescu,˘˘ and L. Grunenfelder,¨ Constructing pointed Hopf algebras by Ore extensions, preprint, 1998. 2. Yu. Bespalov, Hopf algebras: Bimodules, crossproducts, differential calculus, Preprint ITP-93-57E, 1993. 3. Yu. Bespalov, Crossed modules and quantum groups in braided categories, Appl. Categ. Structures 5, No. 2Ž. 1997 , 155᎐204. 4. Yu. Bespalov and B. Drabant, HopfŽ. bi- modules and crossed modules in braided categories, J. Pure Appl. Algebra 123 Ž.1998 , 105᎐129. 5. Yu. Bespalov and B. Drabant, Differential calculus in braided tensor categories, Preprint q-algr9703036, 1997; also, Bicovariant differential calculi and cross products on braided Hopf algebras, in ‘‘Quantum Groups and Quantum Spaces’’Ž. R. Budzynski et al., Eds. , Banach Center Publications, Vol. 40, Institute of Mathematics, Polish Academy of Sciences, Warsaw, 1997. 6. Yu. Bespalov and B. Drabant, Cross product bialgebrasᎏPart II, Preprint DAMTP-98- 117, 1998. 7. R. J. Blattner, M. Cohen, and S. Montgomery, Crossed products and inner actions of Hopf algebras, Trans. Amer. Math. Soc. 298 Ž.1986 , 671᎐711. 8. T. Brzezinski, Crossed products by a coalgebra, Comm. Algebra 25, No. 11Ž. 1997 , 3551᎐3575. 9. S. Caenepeel, B. Ion, G. Militaru, and S. Zhu, Smash biproducts of algebras and coalgebra, preprint, 1998. 10. V. Chari and A. Pressley, ‘‘Quantum Groups,’’ Cambridge Univ. Press, Cambridge, UK, 1994. 11. Y. Doi and M. Takeuchi, Cleft comodule algebras for a bialgebra, Comm. Algebra 14 Ž.1986 , 801᎐818. 12. V. G. Drinfel’d, Quasi-Hopf algebra, Leningrad Math. J. 1 Ž.1990 , 1419᎐1457. 13. P. Freyd and D. Yetter, Braided compact closed categories with applications to low dimensional topology, Ad¨. Math. 77 Ž.1989 , 156᎐182. 14. A. Joyal and R. Street, Braided tensor categories, Macquarie Math. Reports 860081, 1986. 15. A. Joyal and R. Street, Braided tensor categories, Ad¨. Math. 102 Ž.1993 , 20᎐78. 16. C. Kassel, ‘‘Quantum Groups,’’ Graduate Texts in Mathematics, Vol. 155, Springer- Verlag, BerlinrNew York, 1995. CROSS PRODUCT BIALGEBRAS 505
17. G. Lusztig, ‘‘Introduction to Quantum Groups,’’ Progress in Mathematics, Vol. 110, Birkhauser,¨ Boston, 1993. 18. V. V. Lyubashenko, Tangles and Hopf algebras in braided categories, J. Pure Appl. Algebra 98 Ž.1995 , 245. 19. V. V. Lyubashenko, Modular transformations for tensor categories, J. Pure Appl. Algebra 98 Ž.1995 , 279. 20. S. Mac Lane, Natural associativity and commutativity, Rice Uni¨ersity Studies 49 Ž.1963 , 28. 21. S. Mac Lane, ‘‘Categories. For the Working Mathematician,’’ Graduate Texts in Mathe- matics, Vol. 5, Springer-Verlag, BerlinrNew York, 1972. 22. S. Majid, Physics for algebraists: Non-commutative and non-cocommutative Hopf alge- bras by a bicrossproduct construction, J. Algebra 130, No. 1Ž. 1990 , 17᎐64. 23. S. Majid, Braided groups, J. Pure Appl. Algebra 86 Ž.1993 , 187. 24. S. Majid, in ‘‘Cross product quantization, nonabelian cohomology and twisting of Hopf algebrasŽ. V. K. Dobrev, H.-D. Doebner, and A. G. Ushveridze, Eds. , pp. 13᎐41, World Scientific, Singapore, 1995. 25. S. Majid, Some remarks on the quantum double, in ‘‘Proceedings of the 3rd Colloquium on Quantum GroupsŽ. Prague 1994 ,’’ Czech. J. Phys. 44 Ž.1994 , 1059᎐1071.
26. S. Majid, Double-bosonisation of braided groups and the construction of UgqŽ., Math. Proc. Cambridge Philos. Soc., 125 Ž.1999 , 151᎐192. 27. S. Montgomery, ‘‘Hopf Algebras and Their Actions on Rings,’’ CBMS, Vol. 82, Am. Math. Soc., Providence, 1992. 28. D. E. Radford, The structure of Hopf algebras with a projection, J. Algebra 92 Ž.1985 , 322. 29. D. E. Radford, On Kauffman’s knot invariants arising from finite-dimensional Hopf algebras, in ‘‘Advances in Hopf Algebras,’’Ž. J. Bergen and S. Montgomery, Eds. , Lecture Notes in Pure and Applied Mathematics, Vol. 158, pp. 205᎐266, Dekker, New York, 1994. 30. N. Yu. Reshetikhin and V. G. Turaev, Ribbon graphs and their invariants derived from quantum groups, Comm. Math. Phys. 127 Ž.1990 , 1. 31. N. Yu. Reshetikhin and V. G. Turaev, Invariants of 3-manifolds via link polynomials and quantum groups, In¨ent. Math. 103 Ž.1991 , 547. 32. P. Schauenburg, Hopf modules and Yetter᎐Drinfel’d modules, J. Algebra 169 Ž.1994 , 874᎐890. 33. Y. Sommerhauser,¨ Deformed enveloping algebras, New York J. Math. 2 Ž.1996 , 35᎐58. 34. M. E. Sweedler, ‘‘Hopf Algebras,’’ Benjamin, New York, 1969. 35. V. G. Turaev, Quantum invariants of knots and 3-manifolds, Studies in Mathematics, Vol. 18, de Gruyter, BerlinrNew York, 1994. 36. S. L. Woronowicz, Differential calculus on compact matrix pseudogroupsŽ quantum groups.Ž. , Comm. Math. Phys. 122 1989 , 125᎐170. 37. D. Yetter, Quantum groups and representations of monoidal categories, Math. Proc. Cambridge Philos. Soc. 108 Ž.1990 , 261᎐290.