Formality Related to Universal Enveloping Algebras and Study of Hom-(Co)Poisson Algebras Olivier Elchinger

Formality related to universal enveloping algebras and study of Hom-(co)Poisson algebras Olivier Elchinger

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Olivier Elchinger. Formality related to universal enveloping algebras and study of Hom-(co)Poisson algebras. Mathematics [math]. Université de Haute Alsace - Mulhouse, 2012. English. tel-01225555

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HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Public Domain Université de Haute Alsace École Doctorale Jean-Henri Lambert Laboratoire de Mathématiques, Informatique et Applications

Formality related to universal enveloping algebras and study of Hom-(co)Poisson algebras

presented in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy in Mathematics

proposed by

Olivier Elchinger

Thesis directed by Martin Bordemann and Abdenacer Makhlouf defended the 12th of November 2012 in front of the jury composed by:

M. Benjamin Enriquez Université de Strasbourg (Referee) M. Joakim Arnlind University of Linköping (Referee) M. Camille Laurent-Gengoux Université de Metz (Referee) M. Martin Schlichenmaier Université de Luxembourg M. Martin Bordemann Université de Haute Alsace (Advisor) M. Abdenacer Makhlouf Université de Haute Alsace (Advisor) M. Augustin Fruchard Université de Haute Alsace (Invited member)

Remerciements

Je voudrais remercier dans ces lignes les différentes personnes m’ayant soutenu et encouragé durant ces trois années de thèse. Tout d’abord mes parents, qui m’ont suivi toute ma scolarité jusque dans les études supérieures. Mon épouse, qui s’est investie pour moi dans diverses démarches administratives, trajets, et au quotidien ; et me signalait tous les progrès successifs faits par notre fils. Fabien ensuite, qui m’a fait part de l’opportunité d’une place en thèse sur Mulhouse. Takashi et Elodie, ainsi que Philippe, pour leur hospitalité lorsque j’avais besoin de retourner sur Stras- bourg assister à des cours ou des conférences. Je remercie chaleureuse- ment Mme Wurth pour sa disponibilité et son accueil à mon égard. Je tiens aussi à exprimer ma gratitude à Celui qui m’a protégé sur la route plusieurs fois plus longue que prévue. Je souhaite également remercier les membres du laboratoire de m’avoir bien accueilli parmi eux, pour les discussions, énigmes et réflexions menées lors des repas et poursuivies quelques fois plus avancé dans l’après-midi. Merci à Abdennour pour les ré- cents échanges, partages et découvertes. Je remercie vivement Mme Fricker, pour la gestion et résolution d’innombrables détails ad- ministratifs, Mme Robert pour les échanges prompts avec l’école doctorale, ainsi que les personnes des ressources humaines pour leurs conseils. Merci aussi à mes directeurs de thèse pour avoir cru en moi, m’avoir introduit dans leurs recherches, expliqué et ré-expliqué les points moins évidents, fait et refait des calculs gigantesques, envoyé suivre des conférences et des écoles d’été en diverses re- traites et conseillé durant tout ces travaux. Merci de m’avoir offert à de nombreuses reprises le couvert et/ou le café. Je tiens enfin à remercier les rapporteurs et membres du jury d’avoir accepté de prendre le temps de regarder mon travail, et de leurs remarques et critiques pour l’améliorer.

iii iv Contents

Remerciements iii

Contents v

General introduction ix

I Formality and L∞ structures 1

1 Preliminaries 5 1.1 Graded structures ...... 6 1.1.1 Graded vector spaces ...... 6 1.1.2 Shifted spaces ...... 8 1.1.3 Tensor bialgebra ...... 8 1.1.4 Symmetric bialgebra ...... 14 1.1.5 Universal enveloping algebras ...... 17 1.2 Cohomology and deformations ...... 18 1.2.1 Hochschild cohomology ...... 18 1.2.2 Chevalley-Eilenberg cohomology ...... 19 1.2.3 Properties ...... 20 1.2.4 Link with deformations ...... 20

2 Kontsevich formality 23 2.1 Definitions ...... 24 2.2 Case of associative algebras ...... 25 2.2.1 Diﬀerential graded Lie algebras ...... 25 2.2.2 Sections ...... 27 2.2.3 Formality ...... 27 2.2.4 Application to deformation ...... 31 2.3 Case of Lie algebras ...... 32 2.3.1 Polyvectors ﬁelds and polynomials functions . . . 32 2.3.2 Linear Poisson structure ...... 33 2.3.3 Formality ...... 34 2.4 Universal enveloping algebras ...... 35 2.5 Perturbation Lemma ...... 36 2.5.1 Two-degree cohomology ...... 39 2.5.2 General result ...... 42

v 3 Study of the formality for the free algebras 45 3.1 Description of the spaces ...... 46 3.1.1 Deﬁnitions ...... 46 3.1.2 Examples for spaces of dimension 0 and 1 . . . . 47 3.1.3 Results for spaces of dimension greater than 2 . . 48 3.1.4 Case of a ﬁnite dimensional space ...... 55 3.2 Perturbed formality ...... 57 3.2.1 Computations for the cocycle part in the case 1 . . 58 3.2.2 Computations for the cocycle part in the case 2 . . 58 3.2.3 Computations for the coboundary part in the case 1 ...... 59 3.2.4 Computations for the coboundary part in the case 2 ...... 62 4 Study of the formality for the Lie algebra so(3) 65 4.1 Description of the Lie algebra so(3) ...... 66 4.2 Subalgebra of the Chevalley-Eilenberg complex . 67 4.3 Deformation retract of the complex ...... 70 4.4 Computation of the L structure ...... 71 ∞

II Hom-algebraic structures 75

5 Twisting of Hom-(co)algebras 79 5.1 Hom-associative algebras and Hom-Lie algebras . 80 5.1.1 Deﬁnitions ...... 80 5.1.2 Twisting principle ...... 82 5.1.3 Construction of Hom-Lie algebras ...... 83 5.2 Hom-coalgebras, Hom-bialgebras and Hom-Hopf algebras ...... 87 5.2.1 Hom-coalgebras and duality ...... 87 5.2.2 Hom-bialgebra and Hom-Hopf algebra ...... 90 5.3 Hom-Lie coalgebras and Hom-Lie bialgebras . . . . 92

6 Hom-(co)Poisson structures 95 6.1 Hom-Poisson algebras ...... 96 6.1.1 Deﬁnitions and examples ...... 96 6.1.2 Twisting principle ...... 97 6.1.3 Application to Sklyanin algebra ...... 97 6.1.4 Constructing Hom-Poisson algebras from Hom- Lie algebras ...... 98 6.2 1-operation structures ...... 100 6.2.1 Flexibles Hom-algebras ...... 100 6.2.2 Link with Hom-Poisson algebras ...... 101 6.3 Hom-coPoisson algebras and duality ...... 103 vi 6.3.1 Link with Hom-Lie bialgebras ...... 103 6.3.2 Duality ...... 105

7 Deformation and quantization of Hom-algebras109 7.1 Formal Hom-deformation ...... 110 7.1.1 Formal deformation of Hom-associative algebras . 110 7.1.2 Deformations of Hom-coalgebras and Hom-Bialgebras112 7.2 Quantization and twisting of ?-products . . . . . 112 7.2.1 Twists of Moyal-Weyl ?-product ...... 114 7.2.2 Twists of the Poisson bracket ...... 116 7.2.3 Quantization of the Poisson automorphisms . . . 118

A Computations with Mathematica 125 A.1 Computation of the morphisms of sl(2) ...... 126 A.2 Hom-Lie structures associated to the Jackson sl(2) bracket ...... 129

Bibliography 133

vii viii General introduction

his thesis aims to study some algebraic aspects of structures T linked to the problem of deformation quantization. At ﬁrst, we examine the formality for the case of free algebras and for the Lie algebra so(3) and then, we consider deformation quantization for Hom-algebraic structures. The following is about the historic of these subjects, the results are exposed with more details at the beginning of each part.

Deformation of structure theories is useful to formalize quantum physics. If one has a quantum description of a physical system, then the passage to the classical description is done letting “Planck’s constant ~ tend to zero”. The reverse operation, i.e. pro- ducing a quantum description from a classical one is called quantization. The algebraic structure considered by classical mechanics is the associative commutative algebra of smooth functions over a symplectic, or more generally, a Poisson manifold. Deformation quantization consists to construct an associative non-commutative multiplication (more precisely a ?-product) on the formal series in ~ with coefficients in this algebra, which encodes the Poisson bracket in the first order. The Poisson structure is then called the quasi-classical limit and the deformation is the ?-product. This point of view, initiated in 1978 in [BFF+78], tries to consider quantum mechanics as a deformation from classical mechanics, and the Poincaré group as a deformation of the Galileo group. This pioneering work raises the fundamental questions about the ex- istence and the uniqueness of a deformation quantization for a given Poisson manifold. The first results treated the case of symplectics manifolds. The general case of Poisson manifolds was solved by Kontsevich in 1997 in [Kon03]. He deduced the result by proving a much more general statement, which he called “formality conjecture”. En-

ix dowed with the Gerstenhaber bracket, the Hochschild complex of the algebra of smooth functions over a Poisson manifold admits a graded Lie algebraic structure by shift, which controls the deformations of the Poisson bracket. Kontsevich shows that this complex is linked with its cohomology — which therefore controls the same deformations — by a L -quasi-isomorphism, called a formality map. This boils down∞ the problem of deformation to the previously solved case. Konstevich shows in particular that the formality criterion is true for symmetric algebras over a ﬁnite-dimensional vector space are formal. Bordemann and Makhlouf have examined in [BM08] a slight generalization to the case of universal enveloping algebras. They showed (implicit in Kontsevich’s work) that the formality for a Lie algebra is equivalent to those of its universal enveloping algebra. They also proved that there is formality for the universal enveloping algebra of an aﬃne Lie algebra. These methods were used in [BMP05] and give informations about the rigidity of universal enveloping algebras.

In the first part, we will study the question of formality of some classes of Lie algebras. We consider free algebras, a particular case of universal enveloping algebras, and we show that there is no formality in general, except in the trivial cases. The study of the Lie algebra so(3) shows that there is no formality in this case too. The tools used are homological ones. We first recall that the cohomology is concentrated in degrees 0 and 1 for the free algebras and in degrees 0 and 3 for the Lie algebra so(3). We then build a L -quasi-isomorphism between the differential graded Lie algebra of∞ Hochschild’s cochains endowed with the Gerstenhaber bracket and the cohomology endowed with the Schouten bracket. To achieve this, we use a version of the Perturbation lemma adapted to differential graded Lie algebras, which states that given a contraction between two differential graded complex, and a perturbation of one of the differentials, there exists a new contraction between the two complexes endowed with perturbed differentials.

The study of quasi-deformations of Lie algebra of vector fields, in particular the q-deformations of Witt and Virasoro algebra, leads to the introduction of new non-associative structures. Hom-Lie algebras were first introduced by Hartwig, Larsson and Silvestrov in order to describe theses q-deformations using σ-derivations (see [HLS06]). They describe the q-deformation of the Witt algebra by a one-parameter q family of Hom-Lie algebras, such that the initial Witt algebra is obtained for q = 1. The associative type objects x corresponding to Hom-Lie algebras, called Hom-associative algebras, were introduced by Makhlouf and Silvestrov in [MS10b]. The enveloping algebras of Hom-Lie algebras were studied by D. Yau in [Yau08]. The dual notions of Hom-coalgebras, and also Hom- bialgebras, Hom-Hopf algebras and Hom-Lie coalgebras were first studied in [SPAS09, MS10a] and have been enhanced in [Yau11, Yau10a]. The formal deformation theory is extended in [MS10b] to Hom- associatives and Hom-Lie algebras. The theory for bialgebras and Hopf algebras was introduced in [GS92], and was extended to the Hom-coalgebras, Hom-bialgebras and Hom-Hopf algebras in [DM]. Cohomology complexes were build for theses different algebraic structures, and links between cohomology and formal deformation established. The problem of deformation quantization in the Hom-algebraic setting can be formulated as follows: for a given Hom-(co)Poisson algebra, it is to find a deformation of a commutative (co)algebra Hom-(co)associative such that the first term of the deformation corresponds to the initial Hom-(co)Poisson algebra.

The second part of this work was in large part prepublished in the article [BEM12]. In this second part, we recall some properties of Hom-algebraic structures, we introduce the notion of Hom-co- Poisson algebra and study duality. We use a deformation principle by twist to build new structures of the same type, or to deform a classical structure into the corresponding Hom-structure by means of an algebra morphism. In particular, we apply this process to Poisson structures and to ?-products of Moyal-Weyl. We also establish a correspondence between universal enveloping algebras from Hom-Lie algebras endowed with a Hom-coPois- son structure and Hom-Lie bialgebras.

xi xii Part I

Formality and L∞ structures Table of Contents

1 Preliminaries 5 1.1 Graded structures ...... 6 1.2 Cohomology and deformations ...... 18

2 Kontsevich formality 23 2.1 Definitions ...... 24 2.2 Case of associative algebras ...... 25 2.3 Case of Lie algebras ...... 32 2.4 Universal enveloping algebras ...... 35 2.5 Perturbation Lemma ...... 36

3 Study of the formality for the free algebras 45 3.1 Description of the spaces ...... 46 3.2 Perturbed formality ...... 57 4 Study of the formality for the Lie algebra so(3) 65 4.1 Description of the Lie algebra so(3) ...... 66 4.2 Subalgebra of the Chevalley-Eilenberg complex . 67 4.3 Deformation retract of the complex ...... 70 4.4 Computation of the L structure ...... 71 ∞ Table of Contents 3 Introduction of the first part

n this first part, after presenting graded structures in Section 1.1 I and recalling cohomology notions in Section 1.2, we define the notions of L -algebras and formality (Definition 2.1.2 and Defi- nition 2.1.3).∞ A L -algebra is a generalization of a graded Lie algebra. It is some∞ differential graded cocommutative symmetric coalgebra, whose differential encodes a Lie bracket and maps of greater arity. The fact that the differential is of square zero by composition encompass the Jacobi identity at order two and oth- ers identities at any order involving these maps. A L -morphism (Definition 2.1.2) is a map between two L - algebras∞ which intertwine the differentials. Considering the com-∞ ponents of the morphism, this also gives equations at any order.

For an associative algebra, we consider its Hochschild complex and its cohomology which, endowed with the Gerstenhaber bracket and the induced Schouten bracket, become graded Lie algebras by shift. They can be considered as L -algebras, differentials are induced by the brackets. In general,∞ there is no graded Lie algebra morphism injecting the cohomology into the cocycles, but maybe there exists a morphism of L -algebras intertwining the differentials induced by the brackets. ∞ A formality map is such a morphism, if it induce an isomorphism on cohomology (a quasi-isomorphism). If a formality map does not exist, it is always possible to modify order by order the L structure of the cohomology: the differential is then no longer induced∞ by the Schouten bracket only, but contains higher order components. This construction use a version of the Perturbation lemma (Lemma 2.5.4) adapted for differential graded Lie algebras.

We then study the formality equations for free algebras, and show in Section 3.2 that they are not satisﬁed. The computation of the perturbed L structure gives only one more map of arity 3. The same work is done∞ for the Lie algebra so(3). Again, there is no formality (Theorem 4.4.1) and the modiﬁed L structure consists of only one term of arity 3 too. ∞ 4 Table of Contents 1 Preliminaries

Contents 1.1 Graded structures ...... 6 1.1.1 Graded vector spaces ...... 6 1.1.2 Shifted spaces ...... 8 1.1.3 Tensor bialgebra ...... 8 1.1.4 Symmetric bialgebra ...... 14 1.1.5 Universal enveloping algebras ...... 17 1.2 Cohomology and deformations ...... 18 1.2.1 Hochschild cohomology ...... 18 1.2.2 Chevalley-Eilenberg cohomology ...... 19 1.2.3 Properties ...... 20 1.2.4 Link with deformations ...... 20

o begin, we expose the workplace, which is the category of T graded vector spaces. The notion of components shift plays an important role an allows to enrich the involved structures. We then detail some of the structures which can equip tensor spaces. This chapter ends by reminding the notions of Hochschild and Chevalley-Eilenberg cohomology, and some of their properties

5 6 Chapter 1. Preliminaries 1.1 Graded structures

A gradation on a space enable the concise exposition of valid properties in each degree. Computations are also easier by working with homogeneous elements in each component. In the following, K is a ﬁeld of characteristic 0, unless otherwise stated.

1.1.1 Graded vector spaces We consider the category of Z-graded vector spaces: objects are K- ⊕ i i vector spaces Z-graded V = i ZV , direct sum of subspaces V . An element x of V lying in one∈ of the V i is called homogeneous, and we shall denote by i C |x| ∈ Z its degree. In the following, elements will always be homogeneous unless otherwise speciﬁed. Given two graded vector spaces V and W , a linear map φ : V → W is said to be homogeneous of degree j if and only if for all integers i we have φ(V i) ⊂ W i+j. In graded situations we shall write Hom(V,W )j for the vector space of all linear maps which are homogeneous of degree j, and Hom(V,W ) for the direct sum of all the Hom(V,W )j. Clearly, Hom(V,W ) is a graded vector space. Likewise, the tensor product V ⊗ W is graded by setting (V ⊗ i ⊕ k ⊗ i k W ) = k ZV W − , see [ML63]. The tensor product of two mor- ∈ phism φ : V → W and ψ : V 0 → W 0 is deﬁned with the Koszul rule of signs: for all homogeneous elements x ∈ V and y ∈ V 0

ψ x (φ ⊗ ψ)(x ⊗ y) B (−1)| || |φ(x) ⊗ ψ(y), (1.1.1) with ψ of degree |ψ|. There also is the graded transposition

τ : V ⊗ W → W ⊗ V x y (1.1.2) τ(x ⊗ y) B (−1)| || |y ⊗ x. Theses two rules will determine all the signs which will appears in computations. A graded (associative) algebra (A,µ) is a graded vector space A together with graded space morphism µ : A ⊗ A → A of degree 0 satisfying AiAj ⊂ Ai+j.

Very often we write aa0 for µ(a ⊗ a0). For another graded algebra (B,ν) the graded tensor product of A and B is A⊗B with the product map deﬁned as the composite (µ ⊗ ν) ◦ (id ⊗ τ ⊗ id). In terms of elements, the product is given for homogeneous elements a,a0 ∈ A and b,b0 ∈ B by

b a (a ⊗ b)(a0 ⊗ b0) B (−1)| || 0|aa0 ⊗ bb0. 1.1. Graded structures 7

A graded algebra (A,µ) is commutative if µ◦τ = µ, anticommutative if µ ◦ τ = −µ; for a,b ∈ A this writes a b a b ab = (−1)| || |ba et ab = −(−1)| || |ba. A graded coalgebra (C,∆) is defined in a similar way: the comultiplication must satisfy X ∆Cj ⊂ Ck ⊗ Cl. k+l=j It is said cocommutative if τ ◦∆ = ∆ and anticocommutative if τ ◦∆ = −∆. We define a graded coalgebra structure on the tensor product of two graded coalgebras by applying the same rule of signs as in the case for algebras. A derivation of degree i in a graded algebra (A,µ) is a linear morphism d : A → A of degree i such that i a d(ab) = da.b + (−1) | |a.db which also writes d ◦ µ = µ ◦ (d ⊗ id + id ⊗ d). A A A coderivation of degree i in a graded coalgebra (C,∆) is a linear C → C P ⊗ morphism d : of degree i such that if we note ∆a = (a) a1 a2, we have X ⊗ − i a1 ⊗ ∆da = da1 a2 + ( 1) | |a1 da2 (a) or also ∆ ◦ d = (d ⊗ id + id ⊗ d) ◦ ∆. C C Definition 1.1.1 A graded Lie algebra is a Z-graded vector space V endowed with a graded Lie bracket, i.e. a bilinear map [ , ]: V ⊗ V → V such that gradation [V i,V j] ⊂ V i+j and for x,y,z ∈ V graded antisymmetry y x [y,x] = −(−1)| || |[x,y] (1.1.3) graded Jacobi identity

x z y x z y (−1)| || |[[x,y],z] + (−1)| || |[[y,z],x] + (−1)| || |[[z,x],y] = 0 (1.1.4) which also writes x z (−1)| || |[[x,y],z] = 0, x,y,z

where x,y,z indicates a summation on cyclic permutations of x,y,z. 8 Chapter 1. Preliminaries

1.1.2 Shifted spaces For an integer j denote by V [j] the shifted graded vector space defined by V [j]i B V i+j. The identity map V → V induces for each n ∈ Z a map sn : V [j] → V [j − n] which is of degree n because sn(V [j]k) = V j+k = V [j − n]k+n. It is seen as the nth power of the suspension map s B s1 : V [j] → V [j − 1] of degree one. In particular, s : V [1] → V C V [0] and if an element x ∈ V [1] is of degree |x| (in V [1]), then x = sx is of degree |sx| = |x| + 1 in V . The suspension will be ‘visible’ for shifted multilinear maps: let k l φ : V ⊗ → W ⊗ be a multilinear map of degree |φ|. The shifted k → l l j ◦ map φ[j]: V [j]⊗ W [j]⊗ is defined by setting φ[j] B (s⊗ )− k j φ ◦ (s⊗ ) . The degree of the shifted map φ[j] is given by |φ[j]| = k j j(k − l) + |φ| and we have (φ[j])[j0] = φ[j + j0]. Note that (s⊗ ) = k(k 1 j(j 1) − − j k (−1) 2 2 (s )⊗ . In order to compute a shifted map, first write ⊗ ··· ⊗ ∈ k for ξ B x1 xk V ⊗ the value of φ(ξ) with the Sweedler P ⊗···⊗ ∈ notation as φ(1)(ξ) φ(l)(ξ), with φ(i)(ξ) W . By the Koszul ⊗ rule of signs (1.1.1), the value of the shifted map φ[j] on η B y1 ···⊗ ∈ k j ⊗···⊗ j yk V [j]⊗ is computed as follows, with η˜ B s (y1) s (yk):

⊗ ··· ⊗ φ[j](y1 yk) = k(k 1) j(j 1) l(l 1) j( j 1) − − + − − − − j (k 1) y +(k 2) y + +(k (k 1)) y (−1) 2 2 2 2 (−1) − | 1| − | 2| ··· − − | k| X j (l 1) φ (η˜) +(l 2) φ (η˜) + +(l (l 1)) φ (η˜) (−1) − | (1) | − | (2) | ··· − − | (l) | ⊗ ··· ⊗ ⊗ ··· ⊗ ⊗ ··· ⊗ φ(1)(y1 yk) φ(l)(y1 yk).

1.1.3 Tensor bialgebra T ⊕ k For a Z-graded vector space V , we denote by V = k NV ⊗ the ∈ tensorial algebra over V . It is a K-associative graded algebra with unit 1, it inherits its Z-grading by the Z-grading of V , see [ML63]. To avoid any confusion, the symbol ⊗ is not written for the free T multiplication µ = µ V in V , which is the juxtaposition. T T T + ⊕ k Moreover, V is a graded bialgebra: let V = k N? V ⊗ be T →∈ the augmentation ideal. The counit ε = ε V : V K is de- T + T ﬁned by the condition Kerε = V and ε(1) = 1K. The graded comultiplication shuﬄe ∆sh is the morphism of associative algebra T V → T V ⊗ T V induced (by universal property Theorem 1.1.2) ⊗ ⊗ ∈ by its value ∆sh(x) = x 1+1 x on generators x V . Since the multiplication µ[2] on T V ⊗T V is given by (µ⊗µ)◦(id⊗ τ ⊗ id) with the graded transposition, there are signs in formulas ⊗ ⊗ − x y ⊗ involving ∆sh, for example ∆sh(xy) = xy 1+x y + ( 1)| || |y x + x y 1⊗xy, for x ∈ V | | and y ∈ V | |. This comultiplication is graded ◦ cocommutative (i.e. τ ∆sh = ∆sh) of degree 0. 1.1. Graded structures 9

Dualizing this bialgebra structure, we obtain on the space T V another structure of bialgebra, with comultiplication (not graded cocommutative) of deconcatenation, ∆ = ∆ V which dualize the T ··· ⊗ ··· free multiplication and is given by formula ∆(x1 xk) = 1 x1 xk+ Pk ··· ⊗ ··· ··· ⊗ r=2 x1 xr 1 xr xk + x1 xk 1; and multiplication shuﬄe µsh graded commutative,− sometimes written •, which dualize the comultiplication shuﬄe. For an explicit formula of µsh, see equation (1.1.9). The two operations ∆ and µsh are of degree 0, the unit is again 1 and the counit ε. We note prV the canonical projection on T V 1. For two linear maps ψ1,ψ2 from a graded coassociative coalgebra (C,∆C) to a graded associative algebra (A,µA), the convolution ψ1 ? ψ2 of ψ1 and ψ2 with respect to µA and ∆C is given by ◦ ⊗ ◦ ψ1 ? ψ2 B µA (ψ1 ψ2) ∆C. It is a graded associative multiplication in Hom(C,A). T 1 The tensorial algebra ( V ,µ = µ V ,1) is free in the sense that it is characterized up to isomorphismT by the following universal property. → Theorem 1.1.2 Let (A,µA) be an associative graded algebra. Each morphsim φ : V A of graded vector spaces, of degree zero, extends to a unique morphism φ : T V → A of graded algebras.

T φ V / (A,µA) ∈ a ; φ satisfies for each n N ··· ··· (1.1.5) φ φ(x1 xn) = φ(x1) φ(xn) 0 P V ⇔ ◦ (n 1) (n 1) ◦ ⊗ ··· ⊗ φ µ − = µA − (φ φ) Proof. The proof given here presents a construction slightly more explicit than the usual one using the universal property of the tensorial product. ?n (n 1) ◦ n ◦ (n 1) ?0 ?1 Let φ = µA − φ⊗ ∆ − , with φ = 1Aε V and φ = φ, P ?n ⊕T n and let φ = n N φ . Extending φ by 0 on n,1V ⊗ , we have k+1 (k) ··· ∈ φ⊗ ∆ (x1 xn) = 0 for k > n so φ is well-defined. Thus, in ··· the sum defining φ(x1 xn), the only remaining term is exactly ··· φ(x1) φ(xn). So φ is uniquely determined by φ.

The algebra morphism φ induced by φ : V → A is computed as P ?n φ = n N φ , the geometric serie using the convolution with re- ∈ spect to the multiplication µA and the comultiplication of deconcatenation ∆.

Proposition 1.1.3 Let d : V → A be a linear map of degree j ∈ Z. There exists a unique T → ◦ graded derivation of degree j along φ, noted d : V A, i.e. d µA =

1and so often called free algebra 10 Chapter 1. Preliminaries

◦ ⊗ ⊗ | µA (d φ + φ d), such that d V = d.

Proof. Set d(x) B d(x) for x ∈ V , then extend using the formula d(xy) = d(x)φ(y) + φ(x)d(y). Since it respects the associativity of the free multiplication of T V

d(x(yz)) = d(x)φ(yz) + φ(x)d(yz) = d(x)φ(y)φ(z) + φ(x)d(y)φ(z) + φ(x)φ(y)d(z) d((xy)z) = d(xy)φ(z) + φ(xy)d(z) = d(x)φ(y)φ(z) + φ(x)d(y)φ(z) + φ(x)φ(y)d(z)

for all x,y,z ∈ V , the proposition follows.

This derivation induced by d is computed as φ ? d ? φ. In term ··· ∈ T of elements, for x1 xk V , it writes

k ··· X ··· ··· d(x1 xk) = φ(x1) φ(xr 1)d(xr)φ(xr+1) φ(xk). − r=1

A graded coalgebra (C,∆C,εC,1C) is said to be augmented if C is ⊕ + the direct sum C = K1C KerεC. The subspace C B Ker εC is isomorphic to the graded quotient coalgebra C/K1C whitout counit (K1C is a subcoalgebra, thus a coideal of C). A graded coalgebra whitout counit is said to be nilpotent if for each element x ∈ C, there is an integer N such that the N th iteration of the comultiplication vanishes on x. Augmented graded coalgebras (C,∆C,εC,1C) whose C+ (seen as a quotient) are nilpotent form a subcategory C C AN of the category of graded coalgebras. The category AN is closed under tensorial product and contains T V . The coalgebra T C ( V,∆ = ∆ V ,ε) is cofree in the category AN in the sense that it satisﬁes theT following universal property. ∈ C → Theorem 1.1.4 For each coalgebra (C,∆C,εC) AN and each linear map φ : C V of degree 0 vanishing on 1C, there exists a unique morphism of graded → T ◦ coalgebras φ : C V such that prV φ = φ.

T φ V o (C,∆C) ◦ ⊗ ◦ φ satisﬁes ∆ φ = φ φ ∆C X X (1.1.6) prV φ ⇔ φ(x) ⊗ φ(x) = φ(x ) ⊗ φ(x ) ! ! { 1 2 1 2 V (φ(x)) (x)

?n (n 1) ◦ n ◦ (n 1) ?0 ?1 Proof. Let φ = µ − φ⊗ ∆C − , with φ = 1εC and φ = φ, P ?n ∈ and let φ = n N φ , it is well-deﬁned by nilpotency. For x ∈ ∈ K1C, all equations vanish. For x KerεC, we have, in Sweedler 1.1. Graded structures 11 notation

∆ ◦ φ(x) =1⊗φ(x) + φ(x) ⊗ 1 ⊗ ⊗ ⊗ + 1 φ(x1)φ(x2) + φ(x1) φ(x2) + φ(x1)φ(x2) 1 ··· + ∆(φ(x1)φ(x2)φ(x3)) + ⊗ ◦ ⊗ ⊗ (φ φ) ∆C(x) =1εC(x1) φ(x2) + φ(x1) 1εC(x2) ⊗ + 1εC(φ(x1)) φ(x2)φ(x3) ⊗ + φ(x1) φ(x2) ⊗ ··· + φ(x1)φ(x2) 1εC(φ(x3)) + which are equal because of the properties of the counit εC. For a more rigorous calculus, one can show by induction that for n ∈ N,

n n 1 (n) X (i) (n i) i n i X− (i) (n 1 i) ∆ ◦ µ = (µ ⊗ µ − ) ◦ (id⊗ ⊗ ∆ ⊗ id⊗ − ) − µ ⊗ µ − − i=0 i=0

(whit µ(0) = id) and then compute

◦ ◦ X ?n X ◦ (n 1) ◦ n ◦ (n 1) ∆ φ = ∆ φ = ∆ µ − φ⊗ ∆C − n N n N ∈ ∈ n 1 X X− (i) (n 1 i) i n 1 i = (µ ⊗ µ − − ) ◦ (id⊗ ⊗ ∆ ⊗ id⊗ − − ) n N i=0 ∈ n 2 ! −X− (i) ⊗ (n 2 i) ◦ n ◦ (n 1) µ µ − − φ⊗ ∆C − i=0 since Imφ ⊂ V , ∆ ◦ φ = φ ⊗ 1+1⊗φ n 1 X X− (i) (n 1 i) i n 1 i = (µ ⊗ µ − − ) ◦ (φ⊗ ⊗ (φ ⊗ 1+1⊗φ) ⊗ φ⊗ − − n N i=0 ∈ n 2 ! −X− (i) ⊗ (n 2 i) ◦ n ◦ (n 1) µ µ − − φ⊗ ∆C − i=0 n 1 X X− (i) ◦ i+1 ◦ (i) ⊗ (n 2 i) ◦ n 1 i ◦ (n 2 i) ◦ = µ φ⊗ ∆C µ − − φ⊗ − − ∆C − − ∆C n N i=0 ∈ n 1 X− (i 1) ◦ i ◦ (i 1) ⊗ (n 1 i) ◦ n i ◦ (n 1 i) ◦ + µ − φ⊗ ∆C− µ − − φ⊗ − ∆C − − ∆C i=0 n 2 ! − X− (i) ◦ i+1 ◦ (i) ⊗ (n 2 i) ◦ n 1 i ◦ (n 2 i) ◦ µ φ⊗ ∆C µ − − φ⊗ − − ∆C − − ∆C i=0 n ! X X (i 1) ◦ i ◦ (i 1) ⊗ (n 1 i) ◦ n 1 ◦ (n 1 i) ◦ = µ − φ⊗ ∆C− µ − − φ⊗ − ∆C − − ∆C n N i=0 ∈ 12 Chapter 1. Preliminaries

n ! ! XX ?i ⊗ ?n i ◦ X ?r ⊗ X ?s ◦ = φ φ − ∆C = φ φ ∆C n N i=0 r N s N ⊗∈ ◦ ∈ ∈ = (φ φ) ∆C,

P ?n P so φ = n N φ ﬁts. To prove the uniqueness, denote φ = n N φn, ∈ ⊂ n ∈ with Imφn V ⊗ . We have ⊗ ◦ ◦ ⇒ ◦ ⊗ ◦ ◦ ◦ (φ φ) ∆C = ∆ φ µ (φ φ) ∆C = µ ∆ φ. ◦ ··· ··· Since (µ ∆)(x1 xk) = (k+1)x1 xk, this last equation reads φ?φ = P ∈ n N(n + 1)φn. So for all n N, since φ0 = 1εC, ∈ n n 1 X ⇔ − X− φi ? φn i = (n + 1)φn (n 1)φn = φi ? φn i. − − i=0 i=1

?k Pn By induction, supposing φk = φ for 0 6 k 6 n, nφn+1 = i=1 φi ? Pn ?n+1 ?n+1 ?n+1 φn+1 i = i=1 φ = nφ , so we also have φn+1 = φ and − ?n ∈ P ?n thus φn = φ for all n N. So φ = n N φ is uniquely determined by φ. ∈ The coalgebra morphism φ is said to be coinduced by φ : C+ → V P ?n and is computed as φ = n N φ , the geometric serie using the convolution with respect to∈ the free multiplication µ and the comultiplication ∆C. Two coalgebra morphisms Φ,Ψ : T V ← C are equal if and only if ◦ ◦ there components prV Φ and prV Ψ are equal. For example, µsh is coinduced by pr⊗ε + ε ⊗ pr, see equation (1.1.10).

Proposition 1.1.5 Let d : C+ → V be a linear map of degree j ∈ Z. There exists a unique graded coderivation of degree j along φ, noted d : C → T V , i.e. ∆◦d = ⊗ ⊗ ◦ ◦ (d φ + φ d) ∆C, such that prV d = d. Proof. Using similar arguments, d = φ ? d ? φ ﬁts. Indeed, for x ∈ C+, in Sweedler notation, the following expressions are equal.

◦ ∆ d(x) = ∆ d(x) + d(x1)φ(x2) + φ(x1)d(x2) ··· + d(x1)φ(x2)φ(x3) + φ(x1)d(x2)φ(x3) + φ(x1)φ(x2)d(x3) + ⊗ ⊗ ··· = d(x) 1+1 d(x) + ∆(d(x1)φ(x2)) + ∆(φ(x1)d(x2)) +

⊗ ⊗ ◦ ⊗ ⊗ (d φ + φ d) ∆C(x) = d(x1) 1εC(x2) + 1εC(x1) d(x2) ⊗ ⊗ ⊗ + 1 d(x1)φ(x2) + d(x1) φ(x2) + d(x1)φ(x2) 1 ⊗ ⊗ ⊗ ··· + 1 φ(x1)d(x2) + φ(x1) d(x2) + φ(x1)d(x2) 1+ 1.1. Graded structures 13

As before, the composition with µ

◦ ⊗ ⊗ ◦ ⇒ ◦ ◦ ∆ d = (d φ + φ d) ∆C µ ∆ d = d ? φ + φ ? d

gives the uniqueness of d.

Conversely, any graded coderivation D : C → T V is determined ◦ by its component d B prV D. Moreover, d is determined by its restrictions d d| k for any nonnegative integer k. k B V ⊗ Taking the coalgebra C = (T V,∆) and considering coderiva- T → tions along φ = id V , we note for d1,d2 : V V T

◦ ◦ ◦ ◦ X∞ i ⊗ ⊗ j T → d1 G d2 B d1 d2 = d1 (id V ? d2 ? id V ) = d1 id⊗ d2 id⊗ : V V (1.1.7) T T i,j=0

the graded Gerstenhaber multiplication. So, the graded commutator ◦ − − d1 d2 ◦ [d1,d2] = d1 d2 ( 1)| || |d2 d1 is a coderivation along id V which ◦ − − d1 d2 ◦ T is coinduced by prV [d1,d2] = d1 G d2 ( 1)| || |d2 d1 C [d1,d2]G, the graded Gerstenhaber bracket of d1 and d2. There is then equality between

◦ ◦ ◦ ◦ − − d1 d2 ◦ ◦ d G prV [d1,d2] = d [d1,d2] = (d G d1) G d2 ( 1)| || |(d G d2) G d1 and

◦ ◦ ◦ − − d1 d2 ◦ ◦ d G [d1,d2]G = d G (d1 G d2) ( 1)| || |d G (d2 G d1),

known as the Gerstenhaber identity

◦ ◦ − − d1 d2 ◦ ◦ ◦ (d G d1) G d2 ( 1)| || |(d G d2) G d1 = d G [d1,d2]G. (1.1.8) ◦ T So there is a graded pre-Lie identity for G, so (Hom( V,V ),[ , ]G) is a graded Lie algebra. ◦ The structures G and [ , ]G were ﬁrst deﬁned by Gerstenhaber in [Ger63] for V [1] where the graded space was V = V 0.

Lemma 1.1.6 Let µ : V ⊗V → V be a graded associative multiplication (of degree 0). Then the shifted map d = µ[1] : V [1]⊗V [1] → V [1] is of degree 1, and ◦ the associativity of µ is equivalent to d G d = 0. Proof. We have that

2 d : V [1]⊗ → V [1] 1 a 1 a d(a ⊗ b) = s− ◦ µ ◦ (s ⊗ s)(a ⊗ b) = (−1)| |s− µ(s(a) ⊗ s(b)) = (−1)| |ab, 14 Chapter 1. Preliminaries

and for a,b,c ∈ V [1], ◦ ⊗ ⊗ ◦ ⊗ ⊗ (d G d)(a b c) = (µ[1] µ[1])(a b c) = µ[1](µ[1] ⊗ id + id ⊗ µ[1])(a ⊗ b ⊗ c) a = µ[1](µ[1](a ⊗ b) ⊗ c) + (−1)| |µ[1](a ⊗ µ[1](b ⊗ c)) a a + b = (−1)| |µ[1](ab ⊗ c) + (−1)| | | |µ[1](a ⊗ bc) a + ab 2 a + b = (−1)| | | |(ab)c + (−1) | | | |a(bc) 1 since |ab| = |s− (sasb)| = |a| + |b| + 1 2 a + b = (−1) | | | | a(bc) − (ab)c − b 1 = ( 1)| |− asµ[1](a,b,c) = 0.

1.1.4 Symmetric bialgebra S ⊕ Sn The graded symmetric bialgebra on V , V = n N V is defined as the quotient of the free algebra T V by the two-sided∈ ideal gener- x y x y ated by all elements xy − (−1)| || |yx in T V with x ∈ V | |, y ∈ V | |. The resulting associative multiplication, the shuffle multiplication • is graded commutative, i.e. for two homogenous elements a,b ∈ a b S V we have a • b = (−1)| || |b • a, and has for unit element 1. So the quotient S V is a graded commutative associative algebra. Moreover, the first comultiplication ∆sh factors through the quotient and define on S V a graded cocommutative comultiplica- S tion, also noted ∆sh. The space V is then a graded commutative cocommutative bialgebra. It is the free graded commutative algebra generated by V . For an integer n, a permutation σ from the symmetric group Sn ··· ∈ n σ ··· and ξ = x1 xn V ⊗ , denote ξ = xσ(1) xσ(n) the usual right T action from Sn on V . Defining the graded signature from σ with respect to ξ as x x Y σ(i) + (−1)| σ(i)|| σ(j)|σ(j) ( ··· ) e x1 xn,σ B x x i + (−1)| i || j |j 16iσ(j) .σ σ n there is a graded right action ξ = e(ξ,σ)ξ from Sn on V ⊗ , because e(ξ,στ) = e(ξ,σ)e(ξσ ,τ). Using this action, we can give an explicit formula of the shuffle multiplication. Note Sh(k,n − k) the set of shuffle permuta- ∈ ··· tions, i.e. permutations σ Sn such that σ(1) < < σ(k) and σ(k + 1) < ··· < σ(n). Then 1.1. Graded structures 15

··· • ··· X ··· ··· (x1 xk) (xk+1 xn) = e(x1 xn,σ)xσ(1) xσ(n). (1.1.9) σ 1 Sh(k,n k) − ∈ − { } For example, we have Sh(1,1) = S2 = id,(1 2) , and e(x1x2,id) = 1, − x2 x1 e(x1x2,(1 2)) = ( 1)| || |, so

• − x2 x1 x1 x2 = x1x2 + ( 1)| || |x2x1.

1 2 3 1 2 3 { 1 1} Sh(2,1) = id, 1 3 2 , 2 3 1 = id,(2 3)− ,(1 3 2)− − x3 x2 − x3 ( x1 + x2 ) e(x1x2x3,(23)) = ( 1)| || |, e(x1x2x3,(132)) = ( 1)| | | | | | • − x3 x2 − x3 ( x1 + x2 ) x1x2 x3 = x1x2x3 + ( 1)| || |x1x3x2 + ( 1)| | | | | | x3x1x2

1 2 3 1 2 3 { 1 1} Sh(1,2) = id, 2 1 3 , 3 1 2 = id,(1 2)− ,(1 2 3)− − x2 x1 − x1 ( x2 + x3 ) e(x1x2x3,(12)) = ( 1)| || |, e(x1x2x3,(123)) = ( 1)| | | | | | • − x2 x1 − x1 ( x2 + x3 ) x1 x2x3 = x1x2x3 + ( 1)| || |x2x1x3 + ( 1)| | | | | | x2x3x1

The shuﬄe multiplication can be computed recursively as fol- ∈ ··· ∈ T • ··· ··· low. For λ K and x1 xn V , we have λ x1 xn = λx1 xn = ··· • x1 xn λ and ··· • ··· ··· • ··· x1 xk xk+1 xn = x1 (x2 xk xk+1 xn) − xk+1 ( x1 + + xk ) ··· • ··· + ( 1)| | | | ··· | | xk+1 (x1 xk xk+2 xn). ⊗ ⊗ The shuﬄe multiplication is coinduced by the map prV ε+ε prV . ⊗ ⊗ X ⊗ ⊗ ?n µsh = prV ε + ε prV = (prV ε + ε prV ) n N (1.1.10) ⇔ ◦ ∈⊗ ⊗ prV µsh = prV ε + ε prV ··· ··· Indeed, noting ξ = x1 xk et η = xk+1 xn, we obtain by induction X (n 1) ◦ ⊗ ⊗ n ◦ [2](n 1) ··· ⊗ ··· µ − (prV ε + ε prV )⊗ ∆ − (x1 xk xk+1 xn) n N ∈ X (n 1) ◦ ⊗ ⊗ n ⊗ ⊗ ⊗ = µ − (prV ε + ε prV )⊗ (ξ1 η1 ξremain ηremain) n N ∈ • − η1 ξ • = prV (ξ1)ε(η1)(ξremain ηremain) + ( 1)| || |ε(ξ1)prV (η1)(ξremain ηremain) • − η1 ξ • = prV (ξ1)(ξremain η) + ( 1)| || | prV (η1)(ξ ηremain) ··· • ··· = x1 (x2 xk xk+1 xn) − xk+1 ( x1 + + xk ) ··· • ··· + ( 1)| | | | ··· | | xk+1 (x1 xk xk+2 xn). 16 Chapter 1. Preliminaries

◦ ⊗ The associativity of µsh follows easily: to prove that µsh (µsh ◦ ⊗ id) = µsh (id µsh), it suﬃces to have equality on projections on V . ◦ ◦ ⊗ ⊗ ⊗ ◦ ⊗ prV µsh (µsh id) = (prV ε + ε prV ) (µsh id) ⊗ ⊗ ⊗ ⊗ ⊗ = (prV ε + ε prV ) ε + ε ε prV ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ = prV ε ε + ε prV ε + ε ε prV ⊗ ⊗ ⊗ ⊗ ⊗ = prV ε ε + ε (prV ε + ε prV ) ◦ ◦ ⊗ ⊗ ⊗ ◦ ⊗ prV µsh (id µsh) = (prV ε + ε prV ) (id µsh) Like for the tensor coalgebra, the graded cocommutative coal- S CS gebra V is cofree in the category AN of graded cocommuta- tives augmented coalgebras with C+ nilpotent, and the coinduc- tion of morphisms and coderivations is done as in the diagram T S (1.1.6), with V replaced by V . The shuﬄe multiplication µsh of S V can be used instead of the multiplicatoin µ from T V to obtain another convolution ?˜. Coinduced morphisms and coderivations are computed as φ = e?φ˜ instead of the geometric serie, and d = d?e˜ ?φ˜ instead of φ ? d ? φ. S Taking the coalgebra C = ( V,∆sh) and considering coderiva- S → tions along φ = id V , we note for d1,d2 : V V S ◦ ◦ ◦ S → d1 NR d2 B d1 d2 = d1 (d2?id˜ V ): V V (1.1.11) S ∈ Sr the graded Nijenhuis-Richardson multiplication. For d1 Hom( V,V ) ∈ St and d2 Hom( V,V ), it computes as

◦ •···• (d1 NR d2)(x1 xr+t 1) = − t x x + +dx + + \x + + x X Y − ip 1 i1 ip 1 ip 1 ( 1)| | | | ··· | | ··· | − | ··· | − | (1.1.12) 16i1<

Lemma 1.1.7 Let [ , ]: V ⊗ V → V be a graded Lie bracket (of degree 0). Then the shifted map d = [ , ][1] : V [1] • V [1] → V [1] is symmetric of degree 1, ◦ and the Jacobi identity for [ , ] is equivalent to d NR d = 0. 1.1. Graded structures 17

Proof. We have that

2 d : V [1]• → V [1] 1 a 1 a d(a • b) = s− ◦ [ , ] ◦ (s ⊗ s)(a ⊗ b) = (−1)| |s− [s(a),s(b)] = (−1)| |[a,b].

The map d = [ , ][1] is symmetric of degree 1, because

b b +( a +1)( b +1) d(b • a) = (−1)| |[b,a] = −(−1)| | | | | | [a,b] a b a a b = (−1)| || |(−1)| |[a,b] = (−1)| || |d(a • b),

and for x,y,z ∈ V [1], ◦ • • (d NR d)(x y z) y z x ( y + z ) = d(d(x • y) • z) + (−1)| || |d(d(x • z) • y) + (−1)| | | | | | d(d(y • z) • x) x y z + x = (−1)| |d([x,y] • z) + (−1)| || | | |d([x,z] • y) x ( y + z )+ y + (−1)| | | | | | | |d([y,z] • x)

2 x + y +1 y z +2 x + z +1 = (−1) | | | | [[x,y],z] + (−1)| || | | | | | [[x,z],y] x ( y + z )+2 y + z +1 + (−1)| | | | | | | | | | [[y,z],x]

y +1 y z + z +2+( x +1)( z +1) = (−1)| | [[x,y],z] + (−1)| || | | | | | | | [[z,x],y] x ( y + z )+ z +1 + (−1)| | | | | | | | [[y,z],x]

y +1 ( x +1)( z +1) = (−1)| | (−1) | | | | ( x +1)( z +1) ( y +1)( x +1) (−1) | | | | [[x,y],z] + (−1) | | | | [[y,z],x]

( z +1)( y +1) +(−1) | | | | [[z,x],y] = 0

since [ , ] satisﬁes the graded Jacobi identity in V . (The degree of x ∈ V [1] is |x| + 1 where |x| is the degree of x ∈ V ).

1.1.5 Universal enveloping algebras

Let K be a commutative ring and g be a Lie algebra over K. Re- call that a (left) g-representation of g is a K-module M and a K- homomorphism g ⊗ M → M, x ⊗ a 7→ xa such that x(ya) − y(xa) = [x,y]a. To each Lie algebra g, an associative K-algebra U(g) is associated such that every (left) g-representation may be viewed as (left) U(g)-representation and vice-versa. The algebra U(g) is constructed as follows. 18 Chapter 1. Preliminaries

Let T g be the tensor algebra of the K-module g, M T g = T n g where T n g = g ⊗ g ⊗ ··· ⊗ g (n times). n N ∈ 0 1 In particular T g = K1 and T g = g. The multiplication in T g is the tensor product. Every K-linear map g ⊗ M → M has a unique extension to a a T g-module map T g ⊗ M → M. If g ⊗ M → M is a g-module then the vector space g inside T g is in general not a Lie subalgebra being represented on M. This is remedied if and only if the elements of T g of the form x ⊗ y − y ⊗ x − [x,y] where x,y ∈ g, are sent to 0. Consequently, one is led to introduce the two-sided ideal I generated by the elements x ⊗ y − y ⊗ x − [x,y] where x,y ∈ g. The enveloping algebra U(g) of g is thus defined as T g/I. It follows that g-representations and U(g)-modules may be identified. Recall that every U(g)-bimodule M is a g-module by → − M (x,m) xm mx, denoted by a. { } Assume that g is a free Lie algebra. Let xi be a fixed basis → T → U of g and yi be the image of xi by the map g g (g). We ··· set yI = yi1 yip with I a finite sequence of indices i1,...,ip and ∅ yI = 1 if I = . The Poincaré-Birkhoff-Witt Theorem insures that U the enveloping algebra (g) is generated by the elements yI corresponding to increasing sequences I. We denote by S V the symmetric algebra over a K-module V . If Q ⊂ K, then there exists a canonical bijection between S g and U S U (g) which is a g-module isomorphism between g and (ga) (see [Dix74, pp.78-79] or [LV12, Section 3.6.13]).

1.2 Cohomology and deformations

We brieﬂy review the notions of Hochschild and Chevalley-Eilen- berg cohomology. The formal deformation of rings and algebras was introduced by M. Gerstenhaber in 1964 ([Ger66]). He gave a tool to deform algebraic structures based on formal power series. The interest on deformations has grown with the development of quantum groups related to quantum mechanics ([BFF+78]). Ex- amples of quantum groups may be obtained as Hopf algebra deformations of the enveloping algebra of a Lie algebra.

1.2.1 Hochschild cohomology Let (A,µ) be an associative algebra over the field K. For each positive integer n we define the space of Hochschild cochains of de- M n A M A n M gree n with coefficients in ,CH ( , ) B Hom( ⊗ , ), and 0 A M M A M ⊕ n A M CH ( , ) B . We define CH( , ) B n∞=0 CH ( , ) which is 1.2. Cohomology and deformations 19

n A M { } a Z-graded vector space by setting CH ( , ) B 0 for all strictly negative integers n. ∈ n A M th For f CH ( , ), we deﬁne the n Hochschild coboundary operator by

n (δH f )(a1,...,an+1) = a1f (a2,...,an+1) n X − i + ( 1) f (a1,...,ai 1,aiai+1,ai+2,...,an+1) (1.2.1) − i=1 − n+1 + ( 1) f (a1,...,an)an+1. n A M th For each nonnegative integer n denote by HH ( , ) the n Hochschild cohomology group with coeﬃcients in M,

Ker(δn :C n(A,M) → C n+1(A,M)) Z C n(A,M) H n(A,M) H H H H (1.2.2) H B n 1 A M → n A M B n A M Im(δn 1 :CH − ( , ) CH ( , )) BCH ( , ) − n A M { } where again we set HH ( , ) B 0 for all strictly negative integers n. We also deﬁne A M M n A M A M A M HH( , ) B HH ( , ) B Z CH( , )/BCH( , ). n Z ∈ 1.2.2 Chevalley-Eilenberg cohomology Let g a Lie algebra equipped with a representation ρ : g → Hom(V,V ). The Chevalley-Eilenberg space of cochains of degree n is given by n Vk ∈ ? 0 CCE (g,V ) B Hom( g,V ) for each n N ,CCE (g,V ) B g and n { } CCE (g,V ) B 0 for all strictly negative integers n. ∈ n th For f CCE (g,V ), we deﬁne the n Chevalley-Eilenberg coboundary operator by

n n X − i (δCEf )(x0,...,xn) = ( 1) ρ(xi)f (x0,...,xˆi,...,xn) i=0 (1.2.3) X − i+j + ( 1) f ([xi,xj],x0,...,xˆi,...,xˆj,...,xn) 06i

Ker(δn :C n(g,V ) → C n+1(g,V )) Z C n(g,V ) H n(g,V ) CE CE CE CE (1.2.4) CE B n 1 n 1 → n B n g Im(δCE− :CCE − (g,V ) CCE (g,V )) BCCE ( ,V ) n { } where again we set HCE (g,V ) B 0 for all strictly negative integers n. We also deﬁne M n n n n HCE(g,V ) B HCE (g,V ) B ZCCE(g,V )/BCCE(g,V ). n Z ∈ 20 Chapter 1. Preliminaries

1.2.3 Properties We shall need to compute Hochschild and Chevalley-Eilenberg cohomology of U(g) and g, respectively, wherefore we shall cite the following two Theorems. The classical Theorem due to H.Cartan et S.Eilenberg, ([CE56, pp.277]) gives a link between the Hochschild cohomology of an enveloping algebra with values in an U(g)-bimodule M (in particular M = U(g)) and the Chevalley-Eilenberg cohomology of the Lie algebra with values in the same module.

Theorem 1.2.1 Let g be a ﬁnite dimensional Lie algebra over K. Then n U M ' n M ∀ ∈ HH ( (g), ) HCE (g, a) n N In particular, if Q ⊂ K n U U ' n U ' n S ∀ ∈ HH ( (g), (g)) HCE (g, (ga)) HCE (g, g) n N The Hochschild-Serre Theorem [HS53] gives the following fac- torization of the Chevalley-Eilenberg cohomology groups in the case of a decomposable solvable Lie algebra.

Theorem 1.2.2 Let g = n ⊕ t be a ﬁnite dimensional solvable Lie algebra over K, where n is the largest nilpotent ideal of g and t the supplementary subalge- U bra of n, reductive in g, such that the t-module induced on (g)a is semisimple, then for all nonnegative integers p, we have p U ' X i ⊗ j U t HCE (g, (ga)) HCE (t,K) HCE (n, (ga)) . i+j=p j U t where HCE (n, (ga)) denotes the subspace of t-invariant elements.

1.2.4 Link with deformations Let K[[t]] be the power series ring with coeﬃcients in K. For a K- vector space E we denote by E[[t]] the K[[t]]-module of the power A series with coeﬃcients in E. Let ( ,µ0) be an associative (resp. A Lie) K-algebra, then ( [[t]],µ0) is an associative (resp. Lie) K[[t]]- algebra. A formal deformation of an associative (resp. Lie algebra) A is an associative (resp. Lie) K[[t]]-algebra (A[[t]],µ) such that 2 ··· n ··· µ = µ0 + tµ1 + t µ2 + + t µn + , ∈ A ⊗ A A ∈ A ∧ A A where µn Hom( , ) (resp. µn Hom( , )). Moreover, two deformations (A[[t]],µ) and (A[[t]],µ0) are said to be equivalent if there exists a formal isomorphism ··· n ··· ϕ = ϕ0 + ϕ1t + + ϕnt + , 1.2. Cohomology and deformations 21

A ∈ A A with ϕ0 = id (Identity map on ) and ϕn Hom( , ) such that A 1 ∀ ∈ A µ0(a,b) = ϕt− (µ(ϕ(a),ϕ(b)) a,b . A A A deformation of is called trivial if it is equivalent to ( [[t]],µ0). An associative (resp. Lie) algebra A is said to be rigid if every deformation of A is trivial. There is a relation between formal deformation theory and Hoch- schild cohomology in the case of an associative algebra and Che- valley-Eilenberg cohomology in the case of Lie algebra. We denote n A M th by HH ( , ) the n Hochschild cohomology group of an associa- A M n M tive algebra with values in the bimodule and by HCE (g, ) the nth Chevalley-Eilenberg cohomology group of a Lie algebra A with values in a g-module M. The second Hochschild cohomology group of an associative algebra (resp. Chevalley-Eilenberg cohomology group of a Lie algebra) with values in the algebra may be interpreted as the group of infinitesimal deformations. The rigidity Theorem of Gerstenhaber [Ger66] (resp. of Nijenhuis- Richardson [NR66]) insures that if the 2nd Hochschild cohomol- 2 A A 2 ogy group HH ( , ) (resp. Chevalley-Eilenberg HCE (g,g)) of an associative algebra A (resp. a Lie algebra g) vanishes then the algebra (resp. Lie algebra) is rigid. Therefore the semisimple associative (resp. Lie) algebras are rigid because their second cohomology groups are trivial ([GS86]). The third cohomology group corresponds to the obstructions to extend a deformation of order n to a deformation of order n + 1 ([Ger66], [Ger63] and [NR66]). The rigidity of n-dimensional complex rigid Lie algebras was studied by R. Carles, Y. Diakité, M. Goze and J.M. Ancochea-Ber- mudez. Carles and Diakité established the classification for n 6 7 ([CD84], [Car84]), and Ancochea with Goze did the classification for solvable Lie algebras for n = 8 and some classes ([GAB01]). The classification of associative rigid algebras are known up to n 6 6 (see [GM96]). 22 Chapter 1. Preliminaries 2 Kontsevich formality

Contents 2.1 Definitions ...... 24 2.2 Case of associative algebras ...... 25 2.2.1 Diﬀerential graded Lie algebras ...... 25 2.2.2 Sections ...... 27 2.2.3 Formality ...... 27 2.2.4 Application to deformation ...... 31 2.3 Case of Lie algebras ...... 32 2.3.1 Polyvectors ﬁelds and polynomials functions . . . 32 2.3.2 Linear Poisson structure ...... 33 2.3.3 Formality ...... 34 2.4 Universal enveloping algebras ...... 35 2.5 Perturbation Lemma ...... 36 2.5.1 Two-degree cohomology ...... 39 2.5.2 General result ...... 42

eginning with an associative algebra, its Hochschild complex B can be given a graded Lie algebra structure by shift. It is also the case for its cohomology. The question is to know if one can inject the cohomology in the Hochschild complex as a graded Lie subalgebra. It is not possible in general, a more loosely condition is to have a morphism up to homotopy which leads to the notion of formality. The more spectacular application of formality is to obtain an associative formal deformation of the initial algebra. However, even if there is no formality for the preceding Lie brackets, it is always possible to modify the L structure to build a L -morphism. ∞ ∞ 23 24 Chapter 2. Kontsevich formality 2.1 Definitions

⊕ j Let V = j ZV a Z-graded vector space. ∈ Definition 2.1.1 We call differential graded Lie algebra (or dg-Lie algebra) a triple (V,[ , ],δ) where [ , ]: V ⊗ V → V is a Lie bracket and δ : V → V a degree 1 map, the differential, is a derivation of degree 1 of square zero, i.e. satisfying

◦ ◦ ⊗ ⊗ ◦ δ [ , ] = [ , ] (δ idV + idV δ) δ δ = 0. (2.1.1) We consider V [1], the shifted graded vector space of V .

Definition 2.1.2 A L -structure on V is defined to be a graded coderivation D of S∞(V [1]) of degree 1 and such that D2 = 0 and the restriction | S prV [1] D 0(V [1]) = 0, so D is a differential on (V [1]). The coupleS (V,D) is called an L -algebra. ∞ Let (V,D) be an L -algebra and d B prV D whence D = d in the ∞ sense of (1.1.11). For each strictly positive integer k let dk denote | the restriction dk B d k(V [1]). In particular, a gradedS Lie algebra (V,[ , ]) is considered as a L -algebra, since the coderivation D = d induced by d = [ , ][1] is a∞L -structure by Lemma 1.1.7, projecting on V [1]. ∞

A L -morphism from a L -algebra (V,D) to a L -algebra (V 0,D0) is a morphism∞ of diﬀerential∞ graded coalgebras∞Φ : S(V [1]) → S(V 0[1]), i.e. Φ is a morphism of graded coalgebras intertwining diﬀerentials,

⊗ ◦ (Φ Φ)∆ (V [1]) = ∆ (V [1]) Φ and ΦD = D0Φ. (2.1.2) S S 0 Moreover, we say that Φ is a L -quasi-isomorphism if its first ∞ component Φ1 B Φ V [1] induces an isomorphism in cohomology. In particular, a| graded Lie algebra morphism φ :(V,[ , ]) → (V 0,[ , ]0) defines a L -algebra morphism by Φ = φ such that the ∞ map Φ is equal to its first component Φ1 B Φ V [1] which is nothing else that φ[1]. | Let (V,[ , ],δ) be a dg-Lie algebra. We consider the differential D on S(V [1]) induced by δ + [ , ][1] and the homology H of V for the differential δ. If it is again a graded Lie algebra for the induced bracket [ , ]0, we can consider D0, the differential of S(H[1]) induced by [ , ]0[1]. Definition 2.1.3 We say that the L -algebra (V,D) is formal if there is a L -quasi- ∞ ∞ isomorphisme Φ from (H,D0) to (V,D) such that Φ H[1] is an injection into the cocycles. | 2.2. Case of associative algebras 25

In the following, we specialize the notion of formality for the Hochschild complex of an associative algebra, and the Chevalley- Eilenberg complex of a Lie algebra.

2.2 Case of associative algebras

2.2.1 Differential graded Lie algebras Let (A,µ) be an associative algebra over the field K. We note A B ⊕ n A n ZA the space of Hochschild cochains with values in as de- ∈ n n A A A n A fined in Section 1.2.1. We have A B CH ( , ) = Hom( ⊗ , ) for n ∈ N?, A0 B A and An B {0} for strictly negative integers n. In the same way, we note an the nth Hochschild cohomology group and ⊕ n a = n Za . A useful∈ way to obtain properties of the Hochschild cohomology is to consider the Gerstenhaber multiplication ◦ k A A × l A A → k+l 1 A A G :CH ( , ) CH ( , ) CH − ( , )

which is deﬁned on elements by

◦ (f G g)(a1,...,ak+l 1) = − k (2.2.1) X − (i 1)(l 1) ( 1) − − f (a1,...,ai 1,g(ai,...,ai+l 1),ai+l,...,ak+l 1) − − − i=1 and the Gerstenhaber bracket, given by

◦ − − (k 1)(l 1) ◦ [f ,g]G B f G g ( 1) − − g G f .

Proposition 2.2.1 The shifted space (A[1],[ , ]G) is a graded Lie algebra. G We will denote it by ( ,[ , ]G).

Proof. We have

M n M n A[1] = Hom(A[1]⊗ ,A[1]) ' Hom A[1]⊗ ,A[1] = Hom(T V,V ) n N n N ∈ ∈ where V is the underlying vector space of A[1], and using the Ger- T stenhaber identity (1.1.8), we know that (Hom( V,V ),[ , ]G) is a graded Lie algebra. The multiplication µ is an element of A2, seen as m = µ[1] ∈ G1, with |m| = 1. According to Lemma 1.1.6, µ is associative if and only if [m,m]G = 0. 26 Chapter 2. Kontsevich formality

G → G ∈ k Deﬁne b : by b B [m, ]G. For f A , b(f ) = [m,f ]G = ◦ − − f ◦ ∈ A | | − µ[1] G f ( 1)| |f G µ[1]. For a1,...,ak+1 [1] ( ai = 1), ◦ (µ[1] G f )(a1,...,ak+1) ⊗ ⊗ = µ[1](f id + id f )(a1,...,ak+1) ⊗ − a1 f ⊗ = µ[1](f (a1,...,ak) ak+1) + ( 1)| || |µ[1](a1 f (a2,...,ak+1) − k − = ( 1) a1f (a2,...,ak+1) f (a1,...,ak)ak+1 and ◦ ◦ (f G µ[1])(a1,...,ak+1) = (f µ[1])(a1,...,ak+1) k X − i = ( 1) f (a1,...,ai 1,aiai+1,ai+2,...,ak+1), − i=1

− k ◦ ◦ so (δH f )(a1,...,ak+1) = ( 1) µ[1] G f + f G µ[1] (a1,...,ak+1) and

◦ − − k 1 ◦ − f − f δH f = f Gµ[1] ( 1) − µ[1] Gf = [f ,µ[1]]G = ( 1)| |[µ[1],f ] = ( 1)| |b(f ),

that is, the Hochschild coboundary operator δH and b are equal f up to the global sign (−1)| |. Therefore, we can work with b rather than δH .

The operator b is a graded derivation of G. Indeed, for f ,g ∈ G the graded Jacobi identity gives

− µ[1] g − g f − f µ[1] ( 1)| || |[µ[1],[f ,g]G]G + ( 1)| || |[g,[µ[1],f ]G]G + ( 1)| || |[f ,[g,µ[1]]G]G = 0 ⇔ − g − g f − f ( 1)| |b([f ,g]G) + ( 1)| || |[g,b(f )]G + ( 1)| |[f ,[g,µ[1]]G]G = 0 ⇔ − g − − g f + g ( f +1) − − f + g 1 ( 1)| |b([f ,g]G) ( 1)| || | | | | | [b(f ),g]G ( 1)| | | || |[f ,b(g)]G = 0 (2.2.2) ⇔ − g − − − f ( 1)| | b([f ,g]G) [b(f ),g]G ( 1)| |[f ,b(g)]G = 0 ⇔ − f b([f ,g]G) = [b(f ),g]G + ( 1)| |[f ,b(g)]G.

To see that the square of operator b vanishes, taking f = m in the previous equation gives

2 − b (g) = [m,[m,g]G]G = [[m,m]G,g]G [m,[m,g]G]G,

and since µ is associative, the ﬁrst right-hand side term vanishes, 2 − so b (g) = [m,[m,g]G]G = [m,[m,g]G]G = 0. This shows that the G space ( ,[ , ]G,b) is a dg-Lie algebra. The graded Jacobi identity implies that the Gerstenhaber bracket descends to a graded Lie bracket [ , ]s on the shifted cohomology space a[1]. This bracket is called the Schouten bracket, and we will denote the space a[1] by g. So (g,[ , ]s) is a graded Lie algebra. 2.2. Case of associative algebras 27

2.2.2 Sections We shall call a section any graded linear injection φ of degree 0 of the Hochschild cohomology g into the subspace of cocycles of G, i.e.

φ : g → G with b ◦ φ = 0 (2.2.3) such that

p ◦ φ = idg (2.2.4) where p denotes the canonical projection of the space of cocycles onto g. By elementary linear algebra this is always possible.

Proposition 2.2.2 The set of all sections is in bijection with the set of all graded vectors space complements H of BG in ZG.

Proof. Let φ be a section. Deﬁne H B Imφ. Since bφ = 0, we have H ⊂ ZG. Let f ∈ ZG. Then

(2.2.4) p(f − φpf ) = pf − pφpf = pf − pf = 0,

hence f − φpf ∈ BG and ZG = H + BG. Let g ∈ H ∩ BG. Then there exists x ∈ g with g = φx and pg = 0, hence 0 = pg = pφx = x, and g = 0. Hence H ∩ BG = {0}, and ZG = H ⊕ BG. On the other hand, let H be a graded vector space complement to BG in ZG, hence ZG = H⊕BG. Then the restriction of p to H is a 1 linear bijection of degree 0 to g. Deﬁne φ˜ : g → H by φ˜ B p| − H and φ : g → G by the composition φ B i ◦ φ˜, where i : H ,→ G is the canonical injection. Clearly bφ = 0H since the imageH of φ is in H ⊂ ZG. Moreover pφ = p ◦ i ◦ φ˜ = p| φ˜ = idg. Now let ψ : g → G H H any other section having image H, i.e. ψ = i ◦ ψ˜. Then idg = pψ = p| ψ˜, and for all h = ψ˜(x) ∈ H, ψp˜ | h = ψp˜ | Hψ˜(x) = ψ˜(x) = h, hence H 1 H H ψ˜ = p| − = φ˜, which implies ψ = φ. H 2.2.3 Formality In general a section φ : g → G will not be a morphism of graded → G Lie algebras (g,[ , ]s) ( ,[ , ]G): by construction one only has for two classes ξ,η ∈ g φ[ξ,η]s = [φ(ξ),φ(η)]G + b φ2(ξ,η) (2.2.5) so φ is a morphism of graded Lie algebras up to a coboundary b φ2(ξ,η) . One may now hope that one can construct a sequence of linear maps φ1 B φ,φ2,φ3,...,φk,... compatibles with the brackets up to higher homotopies. This gives the notion of L -morphism, ∞ 28 Chapter 2. Kontsevich formality

theses applications being seen as components of a graded coderivation of a graded symmetric algebra which links to the Deﬁnition 2.1.2.

We consider the shifted spaces g[1] and G[1]. We note D = G [ , ]G[1] and d = [ , ]s[1] the shifted brackets on and on g, we also have b[1] = b. Thanks to the shift it follows that both [ , ]G[1] and [ , ]s[1] are graded symmetric maps (Lemma 1.1.7), i.e. [ , ]G[1] S2 G → G is a degree 1 map from ( [1]) [1] and [ , ]s[1] is a degree 1 map from S2(g[1]) → g[1]. Let d : S(g[1]) → S(g[1]) be the unique S coderivation of (g[1]) induced by [ , ]s[1], and let b + D be the S G unique coderivation of ( [1]) induced by b + [ , ]G[1].

Proposition 2.2.3 The applications d and b + D are diﬀerentials, equipping respectively (g,d) and (G,b + D) of structures of L -algebras. ∞ Proof. Projecting on g[1], we have ◦ ⇔ ◦ ⇔ ◦ ⇔ ◦ d d = 0 prg[1] d d = 0 d d = 0 d NR d = 0,

so for d to be a diﬀerential is equivalent that the square of d for the Nijenhuis-Richardson multiplication vanishes, which is equivalent by Lemma 1.1.7 to the graded Jacobi identity of the bracket [ , ]s. In the same way, projecting on G[1], we have

◦ ⇔ ◦ b + D b + D = 0 (b + D) NR (b + D) = 0. ◦ As for d, the graded Jacobi identity of [ , ]G gives D NR D = 0, ◦ ◦ ◦ and since b only takes one argument, b NR b = b G b = b b = 0 because we already know that b is a diﬀerential. It remains to ◦ ◦ ∈ G prove that b NR D + D NR b = 0. For f ,g [1], on one hand we ◦ • • − f have (b NR D)(f g) = b(D(f g)) = ( 1)| |b([f ,g]G), and on the other hand,

◦ • • − f g • (D NR b)(f g) = D(b(f ) g) + ( 1)| || |D(b(g) f ) − b(f ) − f g + b(g) = ( 1)| |[b(f ),g]G + ( 1)| || | | |[b(g),f ]G − f +1 − − f g + b(g) +( b(g) +1)( f +1) = ( 1)| | [b(f ),g]G ( 1)| || | | | | | | | [f ,b(g)]G − f +1 = ( 1)| | [b(f ),g]G + [f ,b(g)]G.

Therefore, ◦ ◦ • − f − − − f +1 (b NR D + D NR b)(f g) = ( 1)| | b([f ,g]G) [b(f ),g]G ( 1)| | [f ,b(g)]G = 0,

since b is a graded derivation of G by (2.2.2). 2.2. Case of associative algebras 29

Deﬁnition 2.2.4 The Hochschild complex G associated to the associative algebra (A,µ) is called formal if there is a L -quasi-morphism (morphism of diﬀerential graded coalgebras of degree∞ 0) Φ : S(g[1]) → S(G[1]), i.e.

⊗ ◦ ◦ ◦ ◦ (Φ Φ) ∆ g[1] = ∆ [1] Φ and b + D Φ = Φ d, (2.2.6) S SG such that the restriction Φ1 of Φ to g[1] is a section. The map Φ is called a formality map. ◦ | The restrictions Φk B pr [1] Φ k g[1] for any positive integer k G S are called the Taylor coeﬃcients of Φ and determine Φ. They rem- edy order-by-order the above mentioned failure of the section map Φ1 to be a morphism of graded Lie algebras.

In particular, a graded Lie algebra morphism φ : g → G deﬁnes a formality map Φ = φ[1].

Originally in [Kon03], formality maps were defined from the space g without shift by a family of multilinear applications. This definition is more useful for computations. Before giving it, we need to define the following signs. ∈ ∀ ∈ ∀ Definition 2.2.5 For k N, x1,...,xk+1 g (homogeneous), 1 6 i < j 6 k + 1,

k(k 1) − 2− ij(x1,...,xk+1) = ( 1) · − ( xi + xj )( x1 + + xi 1 )+ xj ( xi+1 + + xj 1 ) − i+j ( 1) | | | | | | ··· | − | | | | | ··· | − | ( 1) , ∀ ∀ ··· and 1 6 a 6 k, 1 6 i1 < < ik 6 k + 1,

a Y \ − xir ( x1 + +dxi1 + + xir 1 + + xir 1 ) ωa(x1,...,xk+1) = ( 1)| | | | ··· | | ··· | − | ··· | − | r=1 (k+1 a)(k a) (k a)( xi + + xi ) i + +i − − ·(−1) − | 1 | ··· | a | (−1) 1 ··· a (−1) 2 .

{ } k → G Proposition 2.2.6 We consider a sequence of linear maps φk k N? , with φk : g⊗ , satisfying ∈

(i) φ1 is a section − (ii) φk is of degree 1 k

(iii) φk is graded antisymmetric, i.e. for all 1 6 i 6 k,

− − xi xi+1 φk(x1,...,xi 1,xi+1,xi,xi+2,...,xk) = ( 1)| || |φk(x1,...,xk). (2.2.7) − ∈ ∀ ∈ and such that for each k N, x1,...,xk+1 g (homogeneous) 30 Chapter 2. Kontsevich formality

X · ··· ··· ··· ij φk([xi,xj]s,x1, ,xbi, ,xbj, ,xk+1) 16i

where ij B ij(x1,...,xk+1) et ωa B ωa(x1,...,xk+1). ?ϕ˜ X Then Φ = e is a formality map, where ϕ = φk[1]. k>1 Remark 2.2.7

Level k = 0 We have 0 = bφ1, which is equation (2.2.3) for sections.

Level k = 1 The equation is − − φ1([x1,x2]s) = bφ2(x1,x2) 1 + −[φ (x ),φ (x )] + (−1) x1 x2 [φ (x ),φ (x )] 2 1 1 2 2 G | || | 1 2 1 1 G which is (2.2.5).

Level k = 2 We obtain the formula

− φ2([x1,x2]s,x3) = bφ3(x1,x2,x3) x ,x ,x 1 2 3 (2.2.9) − x1 + ( 1)| |[φ1(x1),φ2(x2,x3)]G x1,x2,x3

with the convention that signs from the permutations of the

xi are comprised in the notation x1,x2,x3 . 2.2. Case of associative algebras 31

Sketch of the proof. The link with the ﬁrst deﬁnition appears when S → G taking the projection pr [1] : (g[1]) [1] of the second equation G of (2.2.6). With d = d2 B [ , ]s[1] and D = D2 B [ , ]G[1], we get ◦ ◦ ?ϕ˜ ϕ (d2?id˜ (g[1])) = (b + D2) e . S ◦ ◦ Since b = b pr [1] and D2 = D2 pr 2( [1]), this reads G S G ◦ ◦ 1 ◦ ϕ (d2?id˜ (g[1])) = b ϕ + D2 ϕ?ϕ.˜ S 2 •···• ∈ S k+1 Applying this to y1 yk+1 (g[1]), we get X − ( yi + yj )( y1 + + yi 1 )+ yj ( yi+1 + + yj 1 ) ( 1) | | | | | | ··· | − | | | | | ··· | − | 16i

2.2.4 Application to deformation The formal deformation theory of a dg-Lie algebra G associated to A G an associative algebra ( ,µ0) is very simple if is formal, as Kont- sevich has shown in [Kon03]. (We follow here the presentation of ∈ 2 A A 2 0 [BM08, pp 321–322].) Let π HH ( , )[[h]] = a [[h]] = g[1] [[h]]. Suppose that [π,π]s = 0. Then it is always possible to construct a formal associative defor- P r mation µ = µ0 +µ where µ B r∞=1 h µr such that the cohomology ∗ ∗ class [µ1] of µ1 is equal to π. Consider S(g[1])[[h]] and S(G[1])[[h]] as topological bialgebras (with respect to the h-adic topology) with the canonical extension of all the structure maps. Note that the tensor product is no longer algebraic, but given by S(g[1]) ⊗ S(g[1])[[h]]. Let • denote the shuﬄe-multiplication in a graded symmetric algebra. For a general graded vector space V it can be easily seen that the group-like elements of S V [[h]] are no longer exclusively given by 1, but by exponential functions of any primitive elements of degree zero, i.e. hv 0 hπ they take the form e• with v ∈ V [[h]]. The image Φ(e• ) of the 32 Chapter 2. Kontsevich formality

hπ grouplike element e• in S(g[1])[[h]] under the formality map Φ S G µ is a grouplike element in ( [1])[[h]] and thus takes the form e• ∗ ∈ 2 hπ with µ hA [[h]]. Since [π,π]s = 0 it follows that d(e• ) = 0, ∗ hµ and therefore (b + D)(e• ∗ ) = 0. Projecting this last equation to G[1]0[[h]] = A2[[h]], we get the Maurer-Cartan Equation

1 1 0 = bµ + [µ ,µ ]G = [µ0 + µ ,µ0 + µ ]G, ∗ 2 ∗ ∗ 2 ∗ ∗

showing the associativity of µ = µ0 + µ . Hence µ B µ0 + µ is a ∗ A ∗ formal associative deformation of the algebra ( ,µ0).

2.3 Case of Lie algebras

2.3.1 Polyvectors fields and polynomials functions We consider polyvectors fields as polynomials functions, we re- produce here the explanation of [BM08, pp. 324–325] of this cor- respondance. Let E be a finite dimensional K-vector space which is ungraded (Z-graded of degree 0), and A B S E its symmetric algebra. Ac- cording to the Cartan-Eilenberg Theorem 1.2.1, for each n ∈ N,

n A A ^n ⊗ S T n HH ( , ) E∗ E C poly

where the latter space is the space of algebraic polyvector ﬁelds of T L T n rank n. We set poly = n∞=0 poly.

For the computations, we shall use the canonical identiﬁcation of S E with the algebra of all polynomial functions on the dual { }n { i}n space E∗. Let ei i=1 be a basis of E, and let e i=1 be the dual basis. ∈ S We can regard any f E as a polynomial in the coordinates xi ∈ Pn i (upon writing each x E∗ as x = xie ). The polyvector ﬁelds i=1 V are now polynomial functions with values in E∗. Using partial S derivatives ∂i B ∂/∂xi in E and interior products ıe with respect V i to the dual basis in E∗, we can write the projected Gerstenhaber bracket [ , ]s in its classical form as an algebraic Schouten bracket

n n − ξ 1 X ∧ − − η 1 X ∧ [ξ,η]s B ( 1)| |− ıei ξ ∂iη ( 1)| |− ıei η ∂iξ, (2.3.1) i=1 i=1 where |ξ| et |η| are the ranks of the polyvectors ﬁelds ξ and η in T poly. As the Gerstenhaber bracket, the Schouten bracket deﬁnes a T graded Lie bracket on the shifted space poly[1]. Therefore we have 2.3. Case of Lie algebras 33

∈ T the identities from the Deﬁnition 1.1.1: with ξ,η,ζ poly[1] we have

− − η ξ [η,ξ]s = ( 1)| || |[ξ,η]s, − ξ ζ − η ξ − ζ η ( 1)| || |[[ξ,η]s,ζ]s + ( 1)| || |[[η,ζ]s,ξ]s + ( 1)| || |[[ζ,ξ]s,η]s = 0, T degrees being taken in poly[1]. On the other hand there is the pointwise exterior multiplica- ∧ T S ⊕ − tion which makes poly = (E∗ E∗[ 1]) a graded commutative algebra. The Schouten bracket and the exterior multiplication are compatible by the graded Leibniz rule

T Throughout the computations, we will use the fact that poly is S ∈ T ∈ S ∧ an E-module i.e. for ξ poly, f E, f ξ = f ξ. We also give some examples of Schouten brackets which will be used later. T 1 Example 2.3.1 The space poly is the space of vectors ﬁelds, i.e. the derivations of S T 0 ∈ S ∈ T 1 E = poly. For f ,g E,X,Y poly

[f ,g]s = 0, n X [X,f ]s = X(f ) = Xi∂if , i=1 n n X − X [X,Y ]s = [X,Y ] = Xi∂iY Yi∂iX, i=1 i=1 with in particular − [f ∂i,g∂j] = f (∂ig)∂j g(∂jf )∂i.

2.3.2 Linear Poisson structure

Let E = (g,[ , ]) be a ﬁnite n-dimensional Lie algebra and g∗ its algebraic dual. The Lie algebra structure [ , ] of g can be used to deﬁne a Poisson bracket on g∗ : for f ,g ∈ S g and x ∈ g∗

{f ,g}(x) B x df (x),dg(x) , (2.3.3) which also writes 1 X {f ,g}(x) = P (df ,dg)(x) = Ck x ∂ f ∧ ∂ g 2 ij k i j 16i,j,k6n 34 Chapter 2. Kontsevich formality

∈ S1 ⊗ V2 T 2 with P g g∗ = poly the bivector deﬁned by

n 1 X X P = P ij∂ ∧ ∂ where P ij(x) = Ck x (2.3.4) 2 i j ij k 16i,j6n k=1

k and Cij the structure constants of g. Proposition 2.3.2 With the preceding notations, (S g,·,{ , }) is a Poisson algebra.

Proof. The bracket defined by P satisfies the Leibniz identity because of the Leibniz rule for derivations. The Jacobi relation for P { } is obtained recursively, using the fact that ei,ej = [ei,ej] and that the bracket [ , ] satisfies the Jacobi identity by definition.

2.3.3 Formality

The fact that P is a Poisson structure is equivalent to [P,P ]s = 0. Thus, the operator deﬁnined by δP B [P, ]s is a diﬀerential on the T complex poly, whose cohomology is called Poisson cohomology.

T By computation, the complex ( poly,δP ) identiﬁes with the Che- S T valley-Eilenberg complex (CCE(g, g),δCE). We note A[1] B poly[1] S this diﬀerential graded Lie algebra and a[1] B HCE(g, g)[1] its cohomology with values in S g, which is again a graded Lie algebra, with the induced Schouten bracket [ , ]s0 .

As in the case (Section 2.2.3) where the initial algebra is associative, we consider d the unique coderivation of S(a[2]) induced S by [ , ]s0 [1] and δCE + D, the unique coderivation of (A[2]) induced by δCE + [ , ]s[1]. They are also diﬀerentials, which rewrites as follow.

Proposition 2.3.3 The applications d and δCE + D are diﬀerentials, equipping respectively (a[1],d) and (A[1],δCE + D) of structures of L -algebras. ∞ Deﬁnition 2.3.4 The Chevalley-Eilenberg complex A[1] associated to the Lie algebra (g,[ , ]) is said to be formal if there is a L -quasi-isomorphism Φ : S(a[2]) → S(A[2]), i.e. ∞

⊗ ◦ ◦ ◦ ◦ (Φ Φ) ∆ a[2] = ∆ A[2] Φ et δCE + D Φ = Φ d, (2.3.5) S S such that the restriction Φ1 of Φ to a[2] is a section. The map Φ is called a formality map. As we will see, even if there is no formality map between S(a[2]) and S(A[2]), we can always modify the L structure induced by ∞ 2.4. Universal enveloping algebras 35

= = [ ] in = P and by induction build the compo- d d2 , s0 d n>2 dn nents of the diﬀerential d and of the L -morphism Φ = ϕ. They satisfy the equation ∞

◦ ◦ Φ d = δCE + D Φ, (2.3.6)

which writes, after projection on A[2] and using that D = D2 = [ , ]s[1],

◦ ◦ 1 ◦ ϕ (d?id˜ (a[2])) = δCE ϕ + D2 ϕ?ϕ.˜ (2.3.7) S 2 Since = P , evaluating on •···• ∈ S(a[2]) gives d n>2 dn y1 yk+1

k+1 X X νa(y1,...,yk+1) a=2 16i1<

a Y \ − yir ( y1 + +dyi1 + + yir 1 + + yir 1 ) νa(y1,...,yk+1) = ( 1)| | | | ··· | | ··· | − | ··· | − | r=1

for 1 6 a 6 k + 1 only come from the permutation which place at the beginning the elements of indices i1,...,ia. However, care of signs must be taken when working with the shifted − x maps, for example D2(x,y) = ( 1)| |[x,y]s.

2.4 Universal enveloping algebras

Let (g,[ , ]) be a ﬁnite-dimensional Lie algebra and S g its associated Poisson algebra. With the previous notations, we consider the chain complexes below and their associated shifted brackets.

S S S a B HH( g, g) a0 B HCE(g, g) d B [ , ]s[1] d0 B [ , ]s0 [1] S S U U A B CH( g, g) A0 B CH( g, g) D B [ , ]G[1] D B [ , ]G[1] 36 Chapter 2. Kontsevich formality

The spaces (a[1],[ , ]s) and (a0[1],[ , ]s0 ) are graded Lie algebras and (A[1],[ , ]G,b) and (A0[1],[ , ]G,b) are differential graded Lie algebras. The bracket [ , ]G and the differential b B [m, ]G are in boldface to remind that they come from the associative algebra (U g,m), whose multiplication is different from those of (S g,·). In his celebrated article [Kon03], Kontsevich gives an explicit formula for a formality map

Φ :(S(a[2]),d) → (S(A[2]),b + D).

Considering U g as a convergent deformation of S g, Borde- mann et Makhlouf ([BM08][Theorem 6.1]) twist by conjugation this map and obtain a L -morphism ∞ S → S Φ0 :( (a[2]),δCE + d) ( (A0[2]),b + D).

We use here the identiﬁcation between the chain complexes (a T S poly,δP ) and (CCE(g, g),δCE). This enable to show that the formality associated to a Lie algebra g is equivalent to the one for its universal enveloping algebra U g. S Theorem 2.4.1 ([BM08]) The formality of the chain complex CCE(g, g) is equivalent to U U the formality of the chain complex CH( g, g). S Proof. Suppose that the chain complex CCE(g, g) is formal, so there exists a L -quasi-isomorphism ∞ S → S Φ00 :( (a0[2]),d0) ( (a[2]),δCE + d)

such that Φ100 is a section. Composing with Φ0, we obtain a formality map S → S Φ :( (a0[2]),d0) ( (A0[2]),b + D). S U U Since a0 = HCE(g, g) HH( g, g) by the Cartan-Eilenberg The- orem 1.2.1, Φ = Φ0 ◦ Φ00 indeed is a formality map between the symmetric coalgebras constructed on the Hochschild cohomology and chain complex of U g. U U Conversely, if the chain complex CH( g, g) is formal, we have the previous map Φ and we obtain the formality map Φ00 by com- 1 posing with the inverse L -morphism Φ0− . ∞ 2.5 Perturbation Lemma

Let (G,D) and (g,d) be two L -algebras. We don’t necessarily have a L -morphism from (g,d) to∞ (G,D). Nevertheless, we may perturb ∞ S → G the map d in d0 and construct a map φ : (g[1]) [1] such that 2.5. Perturbation Lemma 37

S → S G Φ = φ : (g[1]) ( [1]) is a L -morphism between (g,d0) and (G,D). ∞

The Perturbation Lemma is a result coming from homological perturbation theory which, beginning with a contraction between two chain complexes and a perturbation of one of the diﬀerentials, enable to build a new contraction between the chain complexes endowed with perturbed diﬀerentials. In [Hue11, Hue10], Hueb- schmann deals with the case of dg-Lie and L algebras ; he shows how to construct a new contraction natural in∞ the given structures. This allows to inductively build a formality map. This construction was also done by Bordemann et al. in [BGH+05, Bor] for L structure and for other operads. ∞

More precisely, we have the following results. We ﬁrst de- ﬁne the terms contraction and perturbation, then we state the Per- turbation Lemma. We then proceed to the construction of a L - morphism in the case where cohomology is concentrated only∞ in degrees 0 and 1, following [BGH+05, A.4, Prop. A.3]. Finally, we give the result without particular hypothesis on the cohomology.

Deﬁnition 2.5.1 A contraction consists of two chain complexes (U,dU ) and (V ,dV ) together with chain maps i : U → V , p : V → U, i.e.

◦ ◦ ◦ ◦ dV i = i dU , dU p = p dV (2.5.1a)

and a map h : V → V of degree −1 such that

◦ p i = idU (2.5.1b) − ◦ idV i p = dV h + hdV (2.5.1c) h2 = 0 h ◦ i = 0 p ◦ h = 0. (2.5.1d)

Then p is a surjection called the projection, i is an injection called ◦ the inclusion and h is an homotopy between idV and i p. We sum up equations (2.5.1) with the diagram

i / (U,dU ) o o (V ,dV ) g h . (2.5.2) p

Remark 2.5.2

1. Condition (2.5.1c) implies that the cohomologies of U and V ◦ − − f g ◦ are isomorphic. Denoting by [f ,g] B f g ( 1)| || |g f the graded commutator of two maps, this equation (2.5.1c) also − ◦ rewrites idV i p = [dV ,h]. 38 Chapter 2. Kontsevich formality

2. Equations (2.5.1d) are called side conditions. They are equivalent to the equation hdV h = h and can always be satisfied replacing h by hdV h if needed. → Definition 2.5.3 A perturbation of the differential dV is a morphism δ : V V of 2 ⇔ 2 degree +1 such that (dV + δ) = 0 δ + [δ,dV ] = 0. Lemma 2.5.4 (Perturbation Lemma) Given a filtered contraction

i / (U,dU ) o o (V ,dV ) g h . p

and a perturbation δ of dV , let

1 X − n ˜ 1 X − n ı˜= (idV + hδ)− i = ( hδ) i h = (idV + hδ)− h = ( hδ) h n N n N ∈ ∈ 1 X − n ˜ 1 X − n p˜ = p(idV + δh)− = p( δh) δ = p(idV + δh)− δi = p( δh) δi. n N n N ∈ ∈ If the ﬁltrations on U and V are complete, then the above series con- ˜ verge, δ is a perturbation of dU and there exists a ﬁltered contraction

i˜ ˜ / (U,dU + δ) o o (V ,dV + δ) g h˜ . p˜

This result was first published in [Bro65]. ˜ ˜ Proof. We show that the new maps ı,˜ p,˜ h,dV + δ,dU + δ satisfies equations (2.5.1). Setting id B idV , we remark first that

1 X n X n 1 δ(id + hδ)− = δ(−hδ) = (−δh) δ = (id + δh)− δ, n N n N ∈ ∈ 1 X n X n 1 (id + hδ)− h = (−hδ) h = h(−δh) = h(id + δh)− ; n N n N − ∈ ∈ − dV (id + hδ) + (id [dV ,h])δ = dV + δ hdV δ = (id + hδ)(dV + δ), − − (id + δh)dV + δ(id [dV ,h]) = dV + δ δdV h = (dV + δ)(id + δh),

2 − − using δ = δdV dV δ in the last two equations. So we obtain ◦ ˜ 1 ◦ ◦ ◦ 1 ı˜ (dU + δ) = (id + hδ)− (i dU + i p δ(id + hδ)− i) 1 ◦ − = (id + hδ)− (dV i + (id [dV ,h])δı˜) 1 − = (id + hδ)− dV (id + hδ) + (id [dV ,h])δ ı˜

1 = (id + hδ)− (id + hδ)(dV + δ) ı˜

= (dV + δ)ı˜ 2.5. Perturbation Lemma 39

and similarly ˜ ◦ ◦ 1 ◦ ◦ 1 (dU + δ) p˜ = (dU p + p(id + δh)− δ i p)(id + δh)− ◦ − 1 = (p dV + pδ˜ (id [dV ,h]))(id + δh)− − 1 = p˜ (id + δh)dV + δ(id [dV ,h]) (id + δh)−

1 = p˜ (dV + δ)(id + δh) (id + δh)−

= p˜(dV + δ).

Moreover

1 1 id − ı˜◦ p˜ = id − (id + hδ)− i ◦ p(id + δh)− 1 − − 1 = (id + hδ)− (id + hδ)(id + δh) (id [dV ,h]) (id + δh)−

1 2 1 = (id + hδ)− [h,dV + δ] + hδ h (id + δh)− and ˜ 1 1 [(dV + δ),h] = (id + hδ)− h(dV + δ) + (dV + δ)h(id + δh)− 1 1 = (id + hδ)− h(dV + δ)(id + δh) + (id + hδ)(dV + δ)h (id + δh)− 1 − 1 = (id + hδ)− [h,dV + δ] h[δ,dV ]h (id + δh)− thus we have − ◦ ˜ id ı˜ p˜ = [(dV + δ),h].

Finally, X X p˜ ◦ ı˜= p (−δh)n (−hδ)mi n N m N ∈ ∈ X − n − m = p ( δh) ( hδ) i = idU n+m=p

because of side conditions (2.5.1d). These conditions on h,i,p implies the same conditions on h,˜ ı,˜ p˜.

2 1 1 h˜ = (id + hδ)− h ◦ h(id + δh)− = 0 1 1 1 h˜ ◦ ı˜= (id + hδ)− h ◦ (id + hδ)− i = (id + hδ)− ◦ h ◦ i = 0 1 1 1 p˜ ◦ h˜ = p(id + δh)− ◦ h(id + δh)− = p ◦ h ◦ (id + δh)− = 0

2.5.1 Two-degree cohomology A L n Let ( ,µ) be an associative algebra, A = n∞=0 A the associated space of Hochschild cochains and a its cohomology. 40 Chapter 2. Kontsevich formality

We suppose that the cohomology is concentred only on two degrees a = a0 ⊕ a1. From the shifted point of view G B A[1] and 1⊕ 0 g B a[1], in order to work with Lie algebras, this writes g = g− g .

We use the previous notations d and b + D from Section 2.2 for the diﬀerentials induced by the shifted brackets, and we construct order by order the perturbed L structure d0 on g to obtain a L - G∞ ∞ morphism between (g,d0) and ( ,b + D). First, the following lemma limits the ‘length’ of φ. 1 ⊕ 0 → G − Lemma 2.5.5 On g = g− g , a map φk : g of degree 1 k vanishes for k > 3. 1 ⊕ 0 Proof. Beginning with g = g− g , since φ1 is a map of degree 0 we have that 1 ⊂ G 1 0 0 ⊂ G0 1 φ1(g− ) − = A and φ1(g ) = A . − In the same way, φ2 is a map of degree 1 hence 1 1 ⊂ G 3 2 { } φ2(g− ,g− ) − = A− = 0 1 0 ⊂ G 2 1 { } φ2(g− ,g ) − = A− = 0 0 0 ⊂ G 1 0 φ2(g ,g ) − = A . ∈ − For all k N, φk is a map of degree 1 k hence

i1 ik ⊂ Gi1+ +ik+1 k φk(g ,...,g ) ··· − − ··· − − − and since 1 6 i1,...,ik 6 0, we have i1 + + ik + 1 k 6 1 k 6 2 for k > 3 so φk = 0 for k > 3.

So the map φ is of the form φ = φ1 + φ2.

Theorem 2.5.6 There is a L -structure d0 = d2 + d3 which confers (g,d0) the structure of a L -algebra,∞ and a L -morphism φ : S (g[1]) → S (G[1]) from G∞ ∞ (g,d0) to ( ,b + D), i.e. which satisﬁes b + D φ = φ d0. Moreover, the S 3 → S 3 1 restriction of d3 : (g[1]) g[1] to (g[1]− ) is a 3-cocycle of the 0 1 Chevalley-Eilenberg cohomology of g with values in g− . − 2 | | Proof. We note E B b + D φ φ d0 and F = d0 . Since b + D = 1 | | | | and φ = 0, we have d0 = 1 and d0 = d2 + d3. Since b + D and d0 are graded Lie coderivations and φ is a morphism of Lie coalgebra, E 2 is a Lie coderivation along φ, so E = φ?E˜ . Moreover, b + D = 0 implies that

b + D E + E d0 + φ F = 0. (2.5.3)

We want to construct φ and d0 such that E and F vanish. Pro- jecting on the ground spaces, we have 2.5. Perturbation Lemma 41

⇔ ⇔ E = 0 E B pr [1] E = 0, F = 0 F B prg[1] F = 0. (2.5.4) G >2 Since d0 : S g[1] → G[1], F takes at least three arguments so | | we have F1 B F 1g[1] = 0 and F2 B F 2g[1] = 0. We project theS equation (2.5.3) onSG[1] and then evaluate it on n arguments.

S 2 G → G S 2 → n = 1 Since D = D2 : ( [1]) [1] and d = d2 : (g[1]) g[1] take two arguments, the only remaining term is bE1 = 0, with E1 = bφ1. 1 ⊂ 0 0 ⊂ 1 A 0 Hence φ1(g− ) ZA and φ1(g ) ZA = Der( ), with A = A. → G We choose a linear injection φ1 : g Z such that pφ1 = idg, so E1 = bφ1 = 0. n = 2 We have bE2 + DE2 + Ed2 + φ1 F 2 = 0. As E1 = 0, E2 = E2 and since D = D2 takes two arguments and E2 only gives one, and since d2 gives one argument to E and E1 = 0, it ¨¨| only remains bE2 = 0. The term E2 writes E2 = bφ2 +¨bφ1 2 + | − Dφ1 2 φ1d2. By deﬁnition of D2, d2 and the projection p : G → | − | − | − Z g, p(Dφ1 2 φ1d2) = pDφ1 2 d2 = 0 so Dφ1 2 φ1d2 is a coboundary. We can then choose φ2 such that E2 = 0. The S 2 0 → G 1 A map φ2 : (g ) − = is deﬁned up to a coboundary, 3 − if another map φ20 is taken, we have Kerb χ2 = φ20 φ2 : S 2(g0) → Z(A).

¨¨ ¨¨ n = 3 We have bE3 +¨DE3 +Ë2d3 + φF3 = 0. Again, E1 = E2 = 0 and and F1 = F2 = 0 implie E3 = E3 and F3 = F3. Since there is G ⊕ ⇔ a split Z = Imφ1 Imb, bE3 + φ1F3 = 0 bE3 = φ1F3 = 0. The map φ1 is injective, so F3 = 0. Using the Lemma 2.5.5, ¨¨ − − the term E3 writes E3 = ¨bφ3 + D2φ φ1d3 φ2d2. Define α : S 3 0 → 1 | | − − (g ) g− , α = 1, as α = D2φ φ2d2. As 0 = bE3 = bα A 1 ⊂ 0 bφ1d3 = bα, α takes its value in Z( ) = φ1(g[1]− ) A . So − the relation D2φ φ2d2 C φ1d3 defines the map d3 such that E3 = 0. n > 4 The map E is of degree |E| = 1. Reasoning with degree in 2 ⊕ 1 g[1] = g− g− as in the Lemma 2.5.5, we find En = 0, and also Fn = 0 for n > 4.

In particular, we have 0 = d0d04, so | | | | 0 = (d2 + d3)(d2 + d3) = d2d2 4 +d2d3 4 + d3d2 4 + d3d3 4, | {z } | {z } =0 =0 | | hence d2d3 4 + d3d2 4 = 0.

42 Chapter 2. Kontsevich formality

◦ ◦ Thus we have 0 = d2d3 + d3d2 = d2 NR d3 + d3 NR d2 = [d2,d3]NR.

∈ 1 Computing on arguments yi g[1]− , we have • • • • • • d2d3(y0 y1 y2 y3) =d2(d3?id˜ )(y0 y1 y2 y3) ⊗ ⊗ • • • =d2µsh(d3 id)(pr3 pr)∆sh(y0 y1 y2 y3) • • • − • • • =d2(d3(y0 y1 y2) y3) d2(d3(y0 y1 y3) y2) • • • − • • • + d2(d3(y0 y2 y3) y1) d2(d3(y1 y2 y3) y0), and • • • • • • d3d2(y0 y1 y2 y3) =d3(d2?id˜ )(y0 y1 y2 y3) ⊗ ⊗ • • • =d3µsh(d2 id)(pr2 pr)∆sh(y0 y1 y2 y3) • • • − • • • =d3(d2(y0 y1) y2 y3) d3(d2(y0 y2) y1 y3) • • • • • • + d3(d2(y0 y3) y1 y2) + d3(d2(y1 y2) y0 y3) − • • • • • • d3(d2(y1 y3) y0 y2) + d3(d2(y2 y3) y0 y1). 0 3 → 0 Noting σ B g ∧ g such that d3 = σ[1] and considering the adjoint representation ρ(x) = adx = [x, ]s, we have 3 3 X δCEσ(x0,x1,x2,x3) = ρ(xi)(σ(x0,...,xbi,...,x3)) i=0 X − i+j + ( 1) σ([xi,xj]s,x0,...,xbi,...,xbj,...,x3) 06i

2.5.2 General result The construction of a perturbed diﬀerential to obtain a new contraction between L -algebras is done by induction in [Hue11], Bordemann gives in∞ [Bor] a closed formula.

Theorem 2.5.7 Let (V ,b) be a chain complex and H its cohomology. Suppose that there is a contraction φ1 / H o o (V ,b) g h π1 and that D is a perturbation of b, or equivalently that (V,b + D) is a L -algebra. Then there is a contraction between graded cocommutative∞ symmetric coalgebras

ϕ S / S ( (H[1]),d) o o ( (V [1]),b + D) g η ψ 2.5. Perturbation Lemma 43

◦ ◦ the homotopy η satisfying η b + D η = η. Moreover, setting ϕ1 = φ1[1] and ψ1 = π1[1], the maps d, ϕ and ψ are graded coalgebra morphisms and are determined by the formulas

◦ ◦ 1 ◦ ◦ d = ψ1 (id (V [1]) + D η)− D ϕ1, (2.5.5) S ◦ 1 ◦ ϕ = (id (V [1]) + η D)− ϕ1, (2.5.6) S ◦ ◦ 1 ψ = ψ1 (id (V [1]) + D η)− . (2.5.7) S This contains in particular that ϕ is a L -morphism satisfying ∞ b + D ◦ ϕ = ϕ ◦ d.

The map η is deﬁned as follow. We set W B Imb ⊕ Imh, so V = ⊕ S S ⊗ S •···• ∈ S H W and (V [1]) (H[1]) (W [1]). Let y1 yk (H[1]) •···• ∈ S and w1 wl (W [1]). Then

η : S(H[1]) ⊗ S(W [1]) → S(V [1]) ( •···• • 1 •···• •···• • •···• h(y1 yk l w1 wl) si l , 0, η(y1 yk w1 wl) = (2.5.8) 0 si l = 0,

where h is the coderivation induced by h.

Corollary 2.5.8 Under the hypothesis of the previous theorem, suppose there is another contraction of chain complexes

φ10 / H0 o o (V ,b) g h0 . π10

Then, the two L structures d and d0 on H and H0 are conjugated i.e. ∞ there is an isomorphism of coalgebras S : S(H0[1]) → S(H[1]) such that

1 ◦ ◦ d0 = S− d S. (2.5.9)

Proof. Using restrictions, since ψ01 = ψ10 = π10 [1] induces an iso- → morphism H0[1] H0[1] and ϕ1 = ϕ1 induces an isomorphism → ◦ | H[1] H[1], we have that (ψ0 ϕ) 1 = ψ10 ϕ1 is an isomorphism, ◦ which implies that ψ0 ϕ is invertible using Lemma 2.5.9 below. We have ◦ ◦ ◦ ◦ ◦ ◦ ψ0 ϕ d = ψ0 b + D ϕ = d0 ψ0 ϕ hence ◦ ◦ ◦ ◦ 1 d0 = ψ0 ϕ d (ψ0 ϕ)− ◦ and S B ψ0 ϕ ﬁts. 44 Chapter 2. Kontsevich formality

It remains to prove the lemma about invertibility. S → S Lemma 2.5.9 Let ϕ : (U) (V ) be a graded coalgebra morphism such that ϕ1 : U → V is an isomorphism. Then ϕ is invertible.

X∞ Sk → Proof. Set ϕ = ϕ1 + ϕ0 with ϕ0 = ϕk where ϕk : (U) V . We k=2 can write

˜ ˜ ϕ = e?ϕ = e?(ϕ1+ϕ0) ˜ ˜ = e?ϕ1 ?e˜ ?ϕ0 ?ϕ˜ 0 − = ϕ1?˜ e 1 V ε U +ϕ1, S S | {z } loc. nil. with the ﬁrst term locally nilpotent. We obtain

◦ 1 ?ϕ˜ 0 − ϕ = ϕ1 id U + ϕ1− ϕ1?˜ e 1 V ε U S S S since

1 ◦ 1 1 prU ϕ1− ϕ1 = ϕ1− prV ϕ1 = ϕ1− ϕ1 prU = prU ◦ 1 1 1 prV ϕ1 ϕ1− = ϕ1 prU ϕ1− = ϕ1ϕ1− prV = prV ,

and so

˜ 1 ◦ ?ϕ1− ϕ0 − ϕ = ϕ1 id U + id U ?˜ e 1 U ε U . S S S S Thus, the following map is well-deﬁned,

˜ 1 1 1 ?ϕ1− ϕ0 − − ◦ 1 ϕ− = id U + id U ?˜ e 1 U ε U ϕ1− S S S S X∞ ˜ 1 k − k ?ϕ1− ϕ0 − ◦ 1 = ( 1) id U ?˜ e 1 U ε U ϕ1− , S S S k=0 it is the inverse of ϕ. 3 Study of the formality for the free algebras

Contents 3.1 Description of the spaces ...... 46 3.1.1 Deﬁnitions ...... 46 3.1.2 Examples for spaces of dimension 0 and 1 . . . . 47 3.1.3 Results for spaces of dimension greater than 2 . . 48 3.1.4 Case of a ﬁnite dimensional space ...... 55 3.2 Perturbed formality ...... 57 3.2.1 Computations for the cocycle part in the case 1 . 58 3.2.2 Computations for the cocycle part in the case 2 . 58 3.2.3 Computations for the coboundary part in the case 1 59 3.2.4 Computations for the coboundary part in the case 2 62

e study the formality for free algebras on a vector space. We W first recall its cohomology, and show that there is formality for spaces of dimension 0 or 1. In greater dimension, we expose the formality equations and finally prove that they can not be satisfied. In finite dimension, we describe the cohomology as the kernel of a trace-form map. Computation of a L structure gives a 3-arity component to the differential induced∞ by the Schouten bracket (of arity 2) and induces a L -morphism which only has its first two component non zero. ∞

45 46 Chapter 3. Study of the formality for the free algebras 3.1 Description of the spaces

Let K be a ﬁeld and V be a K-vector space. We consider the free Lie algebra L(V ) generated by V . Its universal enveloping algebra is isomorphic to the free algebra on V (see [Jac62]),

U(L(V )) T V.

We will work directly with the free algebra T V .

3.1.1 Deﬁnitions

We consider the free algebra A B T V generated by V , deﬁned as usual by

M∞ k M∞ k T V B V ⊗ C T V (3.1.1) k=0 k=0 0 k ⊗ ··· ⊗ where V ⊗ = K, and V ⊗ = V V (k times). The Hochschild cohomology of T V with values in a bimodule M can be computed using a free resolution of T V , it was already known in [CE56, Chap. IX p. 181], for a detailed computation see for example [Hof07, Chap. 5, Prop. 5.3.2].

Theorem 3.1.1 For k ∈ N, the kth Hochschild cohomology group of T V is given by  M V {m ∈ M,∀ a ∈ T V ,am = ma} if k = 0,  T B k T M 1 T M HH ( V, ) = HH ( V, ) if k = 1, (3.1.2)  {0} if k > 2.

V Let M = T V be the deﬁning bimodule. Then T V T is the cen- tre of T V ,

(T T V V if dimV = 0 or dimV = 1, V T = 0 K1 = T V if dimV > 2.

Note that for dimV = 0, T V K and for dimV = 1, V = Kx and T V K[x]. Proposition 3.1.2 For the Hochschild cohomology of T V , the 1-cocycles and 1-coboundaries are of the following form.

ZA1(T V,T V ) = Der(T V,T V ) : = {ϕ ∈ Hom(T V,T V ),ϕ(ab) = ϕ(a)b + aϕ(b)} (3.1.3) 3.1. Description of the spaces 47

and

BA1(T V,T V ) = Inder(T V,T V ) : = {ϕ ∈ Hom(T V,T V ),∃ c ∈ T V,∀ a ∈ T V , ϕ(a) = ca − ac}. (3.1.4)

1 Note that for dimV 6 1, BA (T V,T V ) = {0}.

3.1.2 Examples for spaces of dimension 0 and 1

Let A = T {0} = K. Proposition 3.1.3 The Hochschild complex associated to the algebra A is formal with → 0 φ1 = idK : K A K and φk = 0 for k > 2. k k Proof. We have A = Hom(K⊗ ,K) K for all k and [ , ]G = 0, ( id if k = 2r + 1, r ∈ N, b : Ak → Ak+1 = K 0 if k = 2r, r ∈ N

and therefore ( K if k = 0, ak = 0 if k > 1.

Let A = T (Kx) K[x] be the ring of polynomials in one variable.

Proposition 3.1.4 The Hochschild complex associated to the algebra A = K[x] is formal. Proof. We have

a0 = A and a1 = DerA/ InderA = DerA = Hom(K,A) { ∈ A} = f ∂x,f

is the Lie algebra of vector ﬁelds. | | 1 → A A Let φ1 a0 = id and φ1 a1 : a Hom( , ) the function which A associates to f ∂x its derivation. Then we have

φ1([x,y]s) = [φ1(x),φ1(y)]G

and with φk = 0 for k > 2, the Hochschild complex associated to the algebra A = K[x] is formal. 48 Chapter 3. Study of the formality for the free algebras

3.1.3 Results for spaces of dimension greater than 2

We suppose that dimV > 2. We have already computed

0 T T T V 0 HH ( V, V ) = V T = a K1.

We will consider another description of

1 T T T T T T HH ( V, V ) = Der( V, V )/ Inder( V, V ).

Cocycles are derivations of T V , which can be seen as maps from V to T V ,

Der(T V,T V ) Hom(V,T V ). (3.1.5) Indeed, any derivation of T V is uniquely determined by its values on the generating subspace T V 1 = V . On the other hand, each ψ ∈ Hom(V,T V ) uniquely determines a derivation ψ of T V by

ψ(1) B 0 k (3.1.6) ··· X ··· ··· ψ(v1 vk) B v1 vr 1ψ(vr)vr+1 vk. − r=1 Restricted to T V , the diﬀerential b is exactly the adjoint operator,

b : T V → Hom(T V,T V ) 7→ · x adx B [x, ]

where [·,·] is the usual commutator. We consider the map b˜ : T V → Hom(V,T V ), where the adjoint takes values only in V . This allows to see the coboundaries BA1 = Inder(T V,T V ) b˜ T V + as a subspace of Hom(V,T V ) : we take the adjoints derivations coming from elements of T V +, seen as maps from Hom(V,T V ). Notice that b˜ = b. We have obtained the following result.

Theorem 3.1.5 For dimV > 2, the Hochschild cohomology of T V is

a = a0 ⊕ a1 V = T V T ⊕ Der(T V,T V )/ Inder(T V,T V ) + = K1⊕Hom(V,T V )/b˜ T V .

The cohomology of T V concentrates in two levels, so the The- orem 2.5.6 ensures that there is a L -structure on g = a[1] of the ∞ 3.1. Description of the spaces 49

form d2 + d3, with d3 a 3-cocycle of the Chevalley-Eilenberg cohomology of g. In the following, we will show that the Hochschild complex of T V is not formal for the natural L -structure, and ex- ∞ plicit the term d3.

We will work with the space ZA1 = Hom(V,T V ) rather than Der(T V,T V ), it is a sub-Lie algebra of Hom(T V,T V ) with the ∈ T − bracket [ , ]D : for ψ,χ Hom(V, V ), [ψ,χ]D B ψχ χψ, where ψ and χ are derivations of T V like in (3.1.6). Hom(V,T V ) carries an additional Z-grading according to the degree k k+1 Hom(V,T V ) = Hom(V,V ⊗ ),

this grading is auxiliary, no signs are attached. Since we consider the shifted cohomology

1 0 a[1] = a[1]− ⊕ a[1] + = K1 ⊕ Hom(V,T V )/ T V ,

the elements of Hom(V,T V )/ T V + are of degree 0 for the ﬁrst grading ; thus without ambiguity, we can denote by | | the auxiliary grading of Hom(V,T V ). 1 T + T + 1k k BA V also carries the degree of V , BA = V ⊗ for k > 1. We have

1 ? Hom(V,T V )− = Hom(V,K) = V , (3.1.7) 1 BA1− = {0}, Hom(V,T V )0 = Hom(V,V ), (3.1.8) 0 BA1 = {0}.

Here are some example of the Lie bracket of Hom(V,T V ) which will be used later.

1 ? 0 Example 3.1.6 Let α,β ∈ Hom(V,T V )− = V , A,B ∈ Hom(V,T V ) = Hom(V,V ), ψ ∈ Hom(V,T V ), we have

− [α,β]D = 0 because it is of degree 2 ([ , ]D is of degree 0), [α,ψ]D = αψ, − [A,ψ]D = Aψ ψA, (3.1.9) − [A,B]D = AB BA = [A,B]. 50 Chapter 3. Study of the formality for the free algebras

Moreover, since b˜ = ad, for v,x,y ∈ T V , ˜ bv = adv, ˜ [ψ,adv]D = adψ(v) = bψ(v), (3.1.10) ˜ [adx,ady]D = ad[x,y] = b[x,y]. We set 1 ? H− = V , H0 = Hom(V,V ),

k+1 and choose in each Hom(V,V ⊗ ) a complementary subspace to k+1 k 1k the inner derivations, i.e. Hom(V,V ⊗ ) = H ⊕ BA . We denote by H = L Hn the graded complement of A1 = T . Let n> 1 B b V − k+1 → Hk Pk : Hom(V,V ⊗ ) (3.1.11) be the canonical projection, and for k > 1, let

k+1 → k Qk : Hom(V,V ⊗ ) V ⊗ (3.1.12) − be the canonical map such that id Pk = bQk. Set Q 1 = Q0 = 0. Let = P = P . Note that for ∈ H− ([ ] ) = P k> 1 Pk,Q k>1 Qk x,y ,P x,y D − [x,y]s, where x and y are the canonical projection of x and y in the cohomology Hom(V,T V )/ T V +. T ⊕ T In fact, endowed with the bracket [ , ]D, Ared B V Hom(V, V ) T T is a graded Lie subalgebra of the Hochschild complex A B CH( V, V ) and there is a contraction

i ˜ / (Ared,b) o o (A,b) g Q , P because id − P = bQ˜ . Here we can continue without using the The- orem 2.5.7. → Let φ1 : a A be the section according to the previous decom- 0 ⊂ 0 position. Since φ1(a = K1) A and bφ1(1) = 0, it follows that, ∈ T − for all a V , aφ1(1) φ1(1)a = 0 so φ1(1) = 1 because p(1) = 1. To study the formality of the Hochschild complex of T V , we must verify the equation (2.2.8) for 0 6 k 6 2. On level k = 0, we see that the equation (2.2.4) is satisﬁed by construction. On level k = 1, we have the equation (2.2.5) ∀ ∈ ξ1,ξ2 a[1], φ1([ξ1,ξ2]s) = bφ2(ξ1,ξ2) + [φ1(ξ1),φ1(ξ2)]G 0 which vanishes on both sides if one of the ξi is in a = K1. To lighten computations, we will work in H instead of Hom(V,T V )/ T V +. ∈ 1 ∈ H T For ξ1 = x,ξ2 = y a , x,y , the equation in Hom(V, V ) becomes 3.1. Description of the spaces 51

P ([x,y]D) = bφ2(x,y) + [x,y]D (3.1.13) hence

− φ2(x,y) = Q([x,y]D) + q(x,y)1 (3.1.14) where q : H ∧ H → K is arbitrary, since bq(x,y)1 = 0. ∈ H Let us look at level k = 2. Let x1,x2,x3 . If the Hochschild T complex of V is formal, since φ3 = 0 the equation (2.2.8) would become − φ2(P [x1,x2]D,x3) [x1,φ2(x2,x3)]D = 0 x1,x2,x3 However, using the equation (3.1.14), we obtain

− φ2(P [x1,x2]D,x3) x1(φ2(x2,x3)) x1,x2,x3 − = Q([P [x1,x2]D,x3]D) + q(P [x1,x2]D,x3) + x1(Q[x2,x3]D) (3.1.15) x1,x2,x3 ∈ T C T (x1,x2,x3) V ˜ This element is a scalar, to see this, we will compute that b(T (x1,x2,x3)) = 0. We have

˜ ˜ − ˜ b(T (x1,x2,x3)) = b q(P [x1,x2]D,x3) Q([P [x1,x2]D,x3]D) + b(x1(Q[x2,x3]D)) x1,x2,x3 − ˜ = (P id)([P [x1,x2]D,x3]D) + b(x1(Q[x2,x3]D)) x1,x2,x3 because Imq ⊂ Kerb˜ and bQ˜ = id − P . Moreover, since for v ∈ ∈ T ˜ − V , w V , we have bx1(w)(v) = [x1(w),v] = x1([w,v]) [w,x1(v)], we get ˜ ˜ − b(x1(Q[x2,x3]D))(v) = x1(b(Q[x2,x3]D)(v)) b(Q[x2,x3]D)(x1(v)). Thus, ˜ ◦ ˜ − ˜ ◦ b(x1(Q[x2,x3]D)) = x1 bQ[x2,x3]D bQ[x2,x3]D x1 ˜ = [x1,bQ[x2,x3]D]D − = [x1,[x2,x3]D]D [x1,P [x2,x3]D]D. We then have ˜ − bT (x1,x2,x3) = P ([P [x1,x2]D,x3]D) [P [x1,x2]D,x3]D x1,x2,x3 + [x1,[x2,x3]D]D + [P [x2,x3]D,x1]D = [[x1,x2]s,x3]s + [x1,[x2,x3]D]D x1,x2,x3 = 0 52 Chapter 3. Study of the formality for the free algebras

using Jacobi identity for the Lie brackets [ , ]s and [ , ]D, the other terms cancelling by circular permutation. Since T V = T V + ⊕ Kerε where ε is the counit of T V , and that 3 K T (x1,x2,x3) = ε(T (x1,x2,x3)), equation (3.1.15) becomes

− φ2(P [x1,x2]D,x3) x1(φ2(x2,x3)) x1,x2,x3 (3.1.16) = q([x1,x2]s,x3) + ε(x1(Q[x2,x3]D)) . x1,x2,x3

In the equation (3.1.16), the ﬁrsts term in q of the right-hand side are a scalar 3-coboundary of the Chevalley-Eilenberg cohomology of a, up to a minus sign. As Q is a linear map which is homogeneous of degree 0 (for the grading of Hom( T ) = L Hom( k+1)) since Im ⊂ V, V k> 1 V,V ⊗ Q T + = L k, we have that − V k>1 V ⊗ | | | | ε x1(Q[x2,x3]D) = 0 if x1 > 0 or if [x2,x3]D , 1. (3.1.17) Hence, ε x1(Q[x2,x3]D) is not 0 a priori if and only if Cas 1

| | − | | | | x1 = 1 and x2 = 0 and x3 = 1 (3.1.18) | | − | | | | or x1 = 1 and x2 = 1 and x3 = 0

Cas 2

| | − | | − | | x1 = 1 and x2 = 1 and x3 = 2 (3.1.19) | | − | | | | − or x1 = 1 and x2 = 2 and x3 = 1

Deﬁne

σ : a ∧ a ∧ a → K 7→ (x1,x2,x3) ε pr 1(x1)(Q[x2,x3]D) , (3.1.20) x1,x2,x3 −

T → T 1 ? where pr 1 : Hom(V, V ) Hom(V, V )− = V is the canonical projection.− In general, in the equation (3.1.16), the coboundary (δCEq) does not cancel with the term σ. However we have the following property for σ.

Proposition 3.1.7 σ is a scalar 3-cocycle of the Chevalley-Eilenberg cohomology of a. 3.1. Description of the spaces 53

Proof. We have to verify that

0 = (δCEσ)(x0,x1,x2,x3) − − = σ([x0,x1]s,x2,x3) + σ([x0,x2]s,x1,x3) σ([x0,x3]s,x1,x2) − − σ([x1,x2]s,x0,x3) + σ([x1,x3]s,x0,x2) σ([x2,x3]s,x0,x1)

Since σ and [ , ] are of degree 0, so is (δCEσ), hence we must | | | | | | | | have x0 + x1 + x2 + x3 = 0 ; whitout loss of generality, we can | | | | | | | | assume x0 6 x1 6 x2 6 x3 . There are ﬁve cases :

We show that (δCEσ)(x0,x1,x2,x3) vanishes in the third case, the other are similar (except the ﬁrst, which is direct).

We note x0 = α, x1 = β, x2 = A and x3 = ϕ. − − (δCEσ)(α,β,A,ϕ) = σ([α,β]D,A,ϕ) + σ([α,A]D,β,ϕ) σ([α,ϕ]s,β,A) − − σ([β,A]D,α,ϕ) + σ([β,ϕ]s,α,A) σ([A,ϕ]s,α,β)

Using the equations (3.1.9), (3.1.10), (3.1.17) and then the decom- − position [x,y]s = P [x,y]D = [x,y]D bQ[x,y]D, we get

− (δCEσ)(α,β,A,ϕ) = ε αA(Q[β,ϕ]D) + ε β(Q[ϕ,αA]D) ε β(Q[A,[α,ϕ]s]D) − − ε βA(Q[α,ϕ]D) ε α(Q[ϕ,βA]D) + ε α(Q[A,[β,ϕ]s]D) − − ε α(Q[β,[A,ϕ]s]D) ε β(Q[[A,ϕ]s,α]D) = ε αA(Q[β,ϕ]D) + ε β(Q([ϕ,[α,A]D]D + [A,[ϕ,α]D]D + [α,[A,ϕ]D]D)) − ε α(Q([ϕ,[β,A]D]D + [A,[ϕ,β]D]D + [β,[A,ϕ]D]D)) ˜ − ˜ + ε β(Q[A,bQ[α,ϕ]D]D) ε β(Q[α,bQ[A,ϕ]D]D) − ˜ ˜ − ε α(Q[A,bQ[β,ϕ]D]D) + ε α(Q[β,bQ[A,ϕ]D]D) ε βA(Q[α,ϕ]D) .

The second and third terms vanish by the Jacobi relation. More- ∈ T ˜ ˜ − over, for x,y Hom(V, V ), b[x,Q(y)]D = [x,bQ(y)] = [x,(id P )(y)]D 54 Chapter 3. Study of the formality for the free algebras

− which implies that [x,Q(y)]D = Q[x,(id P )(y)]D + ε([x,Q(y)]D). Adding the fact that α(1) = 0 = β(1), we have

− (δCEσ)(α,β,A,ϕ) = ε αA(Q[β,ϕ]D) ε βA(Q[α,ϕ]D) − + ε β([A,Q[α,ϕ]D]D) ε β([α,Q[A,ϕ]D]D) − ε α([A,Q[β,ϕ]D]D) + ε α([β,Q[A,ϕ]D]D) = 0.

Indeed, the projection by Q gives an element of V , so the action of the bracket on elements of degree 0 is the natural evaluation, for example [A,Q[α,ϕ]D]D = AQ[α,ϕ]D, and since [α,β] = 0 the other terms cancel by the Jacobi relation.

1 Proposition 3.1.8 If we choose another graded complement H0 of BA and therefore another section φ10 , the 3-cocycle σ changes by a coboundary. → Proof. Let φ10 : a A be another section associated with the graded complement H0. Taking the canonical projection p of the space of − − ∈ ˜ cocycles onto a, we get p(φ10 φ1) = 0, so φ10 φ1 Imb and there → T + ˜ exists χ : a BA V such that φ10 = φ1 + bχ. Denote by P 0 the 1 ˜ projection of ZA on H0 and Q0 the map such that bQ0 = id − P 0. ◦ ˜ ◦ ˜ ˜ − We have P 0 = φ10 p = (φ1 + bχ) p = P + bχp so bQ0 = id P 0 = ˜ ˜ ˜ ˜ id − P − bχp = bQ − bχp. Then b(Q0 − (Q − χp)) = 0 and Q0 = Q − χp since b˜ is one-to-one on T V +.

∈ ∈ H Let x1,x2,x3 a, so φ1(x1),φ1(x2),φ1(x3) . As before, pr 1 is ? 1 1 − the projection on V = H− = H0− , so we have for the cocycle σ the following expression : σ(x1,x2,x3) = ε pr 1(x1)(Q[φ1(x2),φ1(x3)]D) . x1,x2x3 −

We compute the cocycle σ 0 associated with the graded complement H0. σ 0(x˙1,x˙2,x˙3) B ε pr 1(x˙1)(Q0[φ10 (x˙2),φ10 (x˙3)]D) x1,x2x3 − − ˜ ˜ = ε pr 1(x˙1)((Q χp)[(φ1 + bχ)(x˙2),(φ1 + bχ)(x˙3)]D) x1,x2x3 − − ˜ = ε pr 1(x˙1)((Q χp)([φ1(x˙2),φ1(x˙3)]D + [φ1(x˙2),bχ(x˙3)]D x1,x2x3 − ˜ ˜ ˜ +[bχ(x˙2),φ1(x˙3)]D + [bχ(x˙2),bχ(x˙3)]D))

◦ ˜ By deﬁnition, we have p[φ1(u),φ1(v)]D = [u,v]s ; p b = 0 implies χpb˜ = 0 and with the equations (3.1.9) and (3.1.10), we have 3.1. Description of the spaces 55

− σ 0(x˙1,x˙2,x˙3) =ε pr 1(x˙1)(Q[φ1(x˙2),φ1(x˙3)]D) ε pr 1(x˙1)(χ[x˙2,x˙3]s) − − ˜ − ˜ + ε pr 1(x˙1)(Qb(φ1(x˙2)(χ(x˙3)))) ε pr 1(x˙1)(Qb(φ1(x˙3)(χ(x˙2)))) − ˜ − + ε pr 1(x˙1)(Qb[χ(x˙2),χ(x˙3)]D) + permutation cyclique des xi − ˜ | T + T + | and since Qb = id V + , and [ V , V ] > 2, G|T

σ 0(x˙1,x˙2,x˙3) = σ(x˙1,x˙2,x˙3) − ε pr 1(x˙1)(χ[x˙2,x˙3]s) + ε pr 1(x˙1)(φ1(x˙2)(χ(x˙3))) − − − ε pr 1(x˙1)(φ1(x˙3)(χ(x˙2))) + permutation cyclique des xi − ∧ → − Let ω : a a K deﬁned by ω(x,y) = ε pr 1(x)(χ(y)) ε pr 1(y)(χ(x)) . − − − Since pr 1([x,y]s) = pr 1(x)φ1(y) pr 1(y)φ1(x), we have − − − − σ 0(x1,x2,x3) = σ(x1,x2,x3) (δCEω)(x1,x2,x3).

3.1.4 Case of a ﬁnite dimensional space

We suppose that V is of ﬁnite dimension N > 2. We can describe H, the graded complement of b˜ T V + as a kernel.

{ } { i} ? Let ei 16i6N be a base of V and e 16i6N its dual basis of V . ∈ ∈ n+1 For n N write the applications ϕ Hom(V,V ⊗ ) as

X i0...in ⊗ ··· ⊗ ⊗ j ϕ = ϕj ei0 ein e . j,i0,...,in

For each n ∈ N, we consider the map

n+1 → n Sn : Hom(V,V ⊗ ) V ⊗ 7→ X ji1...in ⊗ ··· ⊗ Sn(ϕ) ϕj ei1 ein . (3.1.21) j,i1,...,in

Let : Hom( T ) → T be the sum P , it is homoge- S V, V V S B n>0 Sn neous of degree 0.

Proposition 3.1.9 Let n ∈ N. ∩ ˜ n { } (i) KerSn bV ⊗ = 0 . n+1 ⊕ ˜ n Hn (ii) For n > 1, Hom(V,V ⊗ ) = KerSn bV ⊗ , so KerSn. 56 Chapter 3. Study of the formality for the free algebras

∈ n ˜ Proof. (i) Let w V ⊗ , we compute, for later use, Sn(bw) = Sn(ad(w)). For all v ∈ V , we have

ad(w)(v) = wv − vw X i1...in i0 ⊗ ··· ⊗ ⊗ − i0 i1...in ⊗ ··· ⊗ = w v ei1 ein ei0 v w ei0 ein i0,...,in X i i i0...in 1 n − 0 i1...in j ⊗ ··· ⊗ = w − δj δj w v ei0 ein , i0,...,in

P ini1...in 1 − i1...in so Sn(ad(w)) = i ,...,i w − Nw , which we can write −1 n ⊗ ··· ⊗ ⊗ Sn(ad(w)) = ζ(w) Nw, were ζ(ei0 ...ein ) B ei1 ein ei0 is a cyclic permutation extended as a linear map. Since ζn = id n , it follows that ζ is diagonalisable over the algebra, with V ⊗ n 1 eigenvalues 1,u,...,u − where u is a primitive root of unity. Now |u| = 1 and N > 2, hence ζ − Nid is invertible with 1 i 1 1 − 1 X∞ 1 (ζ − Nid)− = −N id − ζ = − ζ N N N i=0 1 1 1 n 1 X∞ 1 = − id + ζ + ··· + ζ − N N N n 1 N in − i=0 1 1 1 1 = − id + ζ + ··· + ζn 1 . − 1 n 1 − N 1 N n N N − ∩ ˜ n { } Hence KerSn bV ⊗ = 0 and Sn is surjective. n+1 n+2 ˜ n n (ii) We have dimHom(V,V ⊗ ) = N , dimbV ⊗ = N for n > n+2 − n 1 and dimKerSn = N N since Sn is surjective. Hence ˜ n ⊕ ˜ n dim(KerSn + bV ⊗ ) = dim(KerSn bV ⊗ ) = N n+2 − N n + N n = N n+2 n+1 = dimHom(V,V ⊗ ),

n+1 ⊕ ˜ n Hn ⊕ ˜ n Hn and Hom(V,V ⊗ ) = KerSn bV ⊗ = bV ⊗ , so KerSn.

This shows that H = L Ker is a graded complement to n> 1 Sn b˜ T V + and thus serves as a section.−

Lemma 3.1.10 Let n ∈ N. ∈ ∈ n+1 (i) For all A Hom(V,V ), ϕ Hom(V,V ⊗ ),Sn([A,ϕ]) = A(Sn(ϕ)). ⊂ ∈ In particular, [A,KerSn] KerSn for all n N. 3.2. Perturbed formality 57

∈ ? ∈ n+1 (ii) For all α V and ϕ Hom(V,V ⊗ ), Sn 1([α,ϕ]) = α(Sn(ϕ))+ i ji ...i − P α ϕ 0 2 n e ⊗ ··· ⊗ e . j,i0,i2...,in i0 j i2 in Proof. (i) We have

n ! i0...in X i0 ki1...in X ir i0...ir 1kir+1...in − i0...in k [A,ϕ]j = Ak ϕj + Ak ϕj − ϕk Aj , k r=1

hence

n X j ki ...i XX i ji ...i ki ...i X ji ...i i ...i i1...in 1n r 1 r 1 r+1 n − 1 nk 1 n S([A,ϕ]) = A ϕ + A ϕ − ϕ A = A(S(ϕ)) k j k j k j k k r=1 k

i0...in 1 P ki0...in 1 Pn i0...ir 1kir+1...in 1 (ii) We have [α,ϕ]j − = k αkϕj − + r=1 αkϕj − − hence

n ! i1...in 1 X kji1...in 1 jki1...in 1 X ji1...ir 1kir+1...in 1 Sn 1([α,ϕ]) − = αkϕj − + αkϕj − + αkϕj − − − k,j r=2 X kji1...in 1 i1...in 1 = αkϕj − + α(Sn(ϕ)) − k,j

3.2 Perturbed formality

We consider again the equation (3.1.16). If the Hochschild complex associated with T V was formal, we should ﬁnd an application q such that the 3-cocycle σ cancels with the 3-coboundary δCEq. In that case, the formality equation (2.2.8) would be satis- ﬁed at the level k = 2. However, we will show that this is not the case, meaning that we have the following result.

Theorem 3.2.1 The cohomology class of the cocycle σ does not vanish in g.

Corollary 3.2.2 Let V un vectorial space, dimV > 2. The Hochschild complex of A = T V is not formal. The formality equation stay perturbed by the 3-cocycle σ, so G we only have a L -morphism between (g,d0 = d2 + σ) and ( ,b + D). The end of this∞ section is the proof of theses results. First, the ﬁnite-dimensional case is done mainly by computations; in any 58 Chapter 3. Study of the formality for the free algebras

dimension (greater than 2) an argument of splitting will be used to obtain the same result in general. We suppose that V is of ﬁnite dimension N > 2. With the equation (3.1.17), we have only the two cases (3.1.18) and (3.1.19) to consider. We obtain more precisely conditions on σ and q in each case.

3.2.1 Computations for the cocycle part in the case (3.1.18) ∈ H Lemma 3.2.3 Let x1,x2,x3 . In the case (3.1.18), σ(x1,x2,x3) vanishes. ∈ H 1 ? ∈ H0 Proof. Let x1 = α − = V , x2 = A = Hom(V,V ) and x3 = ϕ ∈ H1. We want to show that

0 = σ(α,A,ϕ) = ε pr 1(α)(Q[A,ϕ]D) + ε pr 1(A)(Q[ϕ,α]D) + ε pr 1(ϕ)(Q[α,A]D) . − − − − ◦ Since [ϕ,α]D = α ϕ has degree 0 and Q0 = 0, we have that Q[ϕ,α]D = 0 and the second term of the sum vanishes. Similary, ◦ − since [α,A]D = α A has degree 1 and Q 1 = 0, Q[α,A]D = 0 and − the third term also vanishes. Now [A,ϕ]D is of degree 1, but ac- ∈ H1 cording to Lemma 3.1.10, [A,ϕ]D KerS1 = , so the ﬁrst term vanishes.

3.2.2 Computations for the cocycle part in the case (3.1.19)

We now look at the term σ(x1,x2,x3) in the case (3.1.19). We note | | − | | ∈ H2 ⊂ x1 = α, x2 = β and x3 = ϕ, with α = 1 = β and ϕ 3 Hom(V,V ⊗ ). Since [α,β]D = 0, we have in this case σ(α,β,ϕ) = ε α(Q[β,ϕ]D) + ε β(Q[ϕ,α]D) (3.2.1) Proposition 3.2.4 We can write σ(α,β,ϕ) under the form

1 X kjl 1 X ljk kjl σ(α,β,ϕ) = (β α − α β )ϕ = α β (ϕ − ϕ ) (3.2.2) N − 1 l k l k j N − 1 l k j j k,j,l k,j,l Proof. Let us compute Q[α,ϕ] : D i0i1 P ki0i1 i0ki1 i0i1k we have [α,ϕ]Dj = k αkϕj + αkϕj + αkϕj and

3.1.10(ii) i1 X kji1 S1([α,ϕ]D) = αkϕj (3.2.3) k,j ∈ H2 because ϕ so S2(ϕ) = 0. We get ˜ i0i1 i0i1 X kri0 i1 − i0 kri1 (bS1([α,ϕ]D))j = ad(S1([α,ϕ]D))j = αkϕr δj δj ϕr , k,r 3.2. Perturbed formality 59

hence X ˜ i1 kri1 − kri1 − i1 S1(bS1([α,ϕ]D)) = αkϕr Nαkϕr = (1 N)S1([α,ϕ]D) . k,r It follows that 1 [α,ϕ] + bS˜ ([α,ϕ] ) = P [α,ϕ] = [α,ϕ] D N − 1 1 D D s H1 is in KerS1 = , and so 1 −1 [α,ϕ] = [α,ϕ] + bS˜ ([α,ϕ] ) + b˜ S ([α,ϕ] ) . D D N − 1 1 D N − 1 1 D Hence

1 (3.2.3) 1 X kji Q[α,ϕ] = − S ([α,ϕ] ) = − α ϕ 1 e , D N − 1 1 D N − 1 k j i1 k,j,i1 and the term (3.2.1) becomes

1 X kjl 1 X ljk kjl σ(α,β,ϕ) = (β α − α β )ϕ = α β (ϕ − ϕ ). N − 1 l k l k j N − 1 l k j j k,j,l k,j,l

3.2.3 Computations for the coboundary part in the case (3.1.18) In the case (3.1.18), the ε part vanishes using the Lemma 3.2.3. For the Hochschild complex associated with T V to be formal, we need that the q part also vanishes, so the arbitrary map q : H ∧ H → K ∈ H must satisfy, for x1,x2,x3 as in the case (3.1.18),

− 0 = (δCEq)(x1,x2,x3) = q([x1,x2]s,x3) + q([x2,x3]s,x1) + q([x3,x1]s,x2). (3.2.4) We have H0 = Hom( ) gl ( ). Note = . As V,V N K qe q glN (K) glN (K) ∈ H0 | ∧ σ(x1,x2,x3) = 0 for xi = glN (K), we already need that (δCEqe) = ∈ 2 0 i.e. that qe ZCE(glN (K),K) is a 2-cocycle of the scalar Chevalley- Eilenberg cohomology of glN (K). nd Proposition 3.2.5 The 2 group of the scalar Chevalley-Eilenberg cohomology of glN (K) 2 is trivial, i.e. HCE (glN (K),K) = 0. { } Proof. Let Eij 16i,j6N be a base of glN (K), we have

N ⊕ X glN (K) = slN (K) wK, where w = Eii. k=1 Using the Hochschild-Serre Theorem 1.2.2, we have that 60 Chapter 3. Study of the formality for the free algebras

2 M 2 k ⊗ 2 k 2 slN (K) HCE (glN (K),K) = HCE (slN (K),K) HCE − (wK,K) HCE(wK,K) = 0, k=0

because  K if k = 0 k  HCE (slN (K),K) = 0 if k = 1  0 if k = 2

using the theorem of Whitehead.

Therefore, the 2-cocylce qe is a 2-coboundary, i.e. there exists p : → ∀ ∈ − glN (K) K such that A,B glN (K), qe(A,B) = p([A,B]). There ∈ ∈ P j i exists P glN (K) such that for A glN (K), p(A) = i,j Pi Aj = tr(PA).

Now note q 1,1 = q 1 1 , we can write it under the form − |H− ∧H

2 ∧ ? → q 1,1 : Hom(V,V ⊗ ) V K −   ! X ij · ⊗ k ∧ X l 7→ X kl ij  ψk ei ej e  αle q¯ij ψk αl k,i,j l i,j,k,l

kl Proposition 3.2.6 Under the condition (3.2.4), we can write q¯ij as

1 q¯kl = P kδl + P kδl − δkP l + νδkδl − Nνδkδl, ij i j j i N i j i j j i

where P = Q + ρ1, tr(Q) = 0 and ν ∈ K.

1 0 1 Proof. For α ∈ H− ,A ∈ H , ψ ∈ H , we have

ij X i rj j ir − r ij [A,ψ]Dk = Arψk + Arψk Akψr r i − i −X ri ir [ψ,α]Dk = [α,ψ]Dk = αrψk + αrψk , r 3.2. Perturbed formality 61

so the equation (3.2.4) reads, with α,A,ψ − 0 = q 1,1(ψ,[α,A]D) q 1,1([A,ψ]D,α) + tr p([[ψ,α]D,A]) − − X kl ij r − kl i rj − kl j ir kl r ij = q¯ij ψk αrAl q¯ij Arψk αl q¯ij Arψk αl + q¯ij Akψr αl i,j,k,l,r ! X k X − ri s − ir s i rs i sr + Pi αrψs Ak αrψs Ak + Asαrψk + Asαrψk i,j,k,l,r s X ij k rl l kr − r kl − r kl = ψk αl Ar q¯ij + Arq¯ij Ai q¯rj Aj q¯ir i,j,k,l,r − l k r − l k r l r k l r k δi Ar Pj δjAr Pi + δi Aj Pr + δjAi Pr X ij kl = ψk αlMij i,j,k,l,r

kl kl k l k l where Mij = [A,q]Dij + [P,A]Dj δi + [P,A]Di δj. ∈ 2 2 Deﬁne p 1,1 Hom(V ⊗ ,V ⊗ ) by − X kl i · j ⊗ · p 1,1 = p¯ij e e ek el − i,j,k,l X k l k l i · j ⊗ · = pi δj + pj δi e e ek el, i,j,k,l

kl − kl so we get Mij = [A,q 1,1 p 1,1)]D . − − ij We can decompose

ij i j ∈ ∈ ψk = Bkv with v V,B glN (K) and tr(B) = 0, ∈ H1 j P kj because ψ = KerS1, so 0 = S1(ψ) = k ψk . Then we have ∀ ∈ ? ∀ ∈ ∀ ∈ α V , v V, B glN (K) with tr(B) = 0 i j kl ⇒ i kl ∀ 0 = Bkv αlMij BkMij = 0 B with tr(B) = 0

kl 1 k P rl In particular, for B = Eij, we get Mij = N δi r Mrj , so we have ∀ ∈ A glN (K) − − 1 − 0 = A,(q 1,1 p 1,1) tr(q 1,1 p 1,1) . − − N − − D Recall the invariant tensor theorem [KMS93, Theorem 24.4 page 214].

k ? l Theorem 3.2.7 Let t ∈ V ⊗ ⊗ V ⊗ X i ...i t = t 1 k e ···e ⊗ ej1 ···ejl j1...jl i1 ik i1,...,ik,j1,...,jl 62 Chapter 3. Study of the formality for the free algebras

∀ ∈ such that A GL(V ), [A,t]D = 0. Then ( 0 if k , l t = P σ S aσ σ if k = l ∈ k ∈ where aσ K and

i ...i σ = σ¯ 1 k e ···e ⊗ ej1 ···ejl j1...jl i1 ik i i 1 ··· k ··· ⊗ j1 ··· jl = δjσ(1) δjσ(k)ei1 eik e e .

This theorem gives us

kl − − 1 − k l k l (q 1,1 p 1,1) tr(q 1,1 p 1,1) = λδi δj + µδj δi , − − N − − ij so

X 1 1 1 q¯kl = P kδl + P kδl + q¯rl − P rδkδl − δkP l + λδkδl + µδkδl. ij i j j i N rj N r i j N i j i j j i r

The contraction k ! i gives µ = −Nλ and as before, since ∀ ψ ∈ H1 ∀ ∈ ? P rl rj P rl = KerS1, α V we have r,j,l q¯rjψr αl = 0, we get r q¯rj = 0. ∈ ∈ Finally, replacing P by Q+ρ1, with ρ K, Q glN (K), tr(Q) = 0, we have 1 ρ q¯kl = Qkδl + Qkδl − δkQl + νδkδl − Nνδkδl with ν = λ − . ij i j j i N i j i j j i N

3.2.4 Computations for the coboundary part in the case (3.1.19) ∈ H Proposition 3.2.8 Let x1,x2,x3 . In the case (3.1.19), (δCEq)(x1,x2,x3) vanishes. | | − | | Proof. We note x1 = α, x2 = β and x3 = ϕ, with α = 1 = β and ∈ H2 ⊂ 3 ϕ Hom(V,V ⊗ ). Since [α,β]D = 0, and considering the degrees, the q part left is

− q 1,1(P [β,ϕ]D,α) q 1,1([α,ϕ]D,β) − − In the Section 3.2.2, we have computed that

1 P [α,ϕ] = [α,ϕ] + bS˜ ([α,ϕ] ), D D N − 1 1 D 3.2. Perturbed formality 63 so we can write − q 1,1(P [β,ϕ]D,α) q 1,1(P [α,ϕ]D,β) − − ! X rk lij ilj ijl 1 X j lsi i lsj = α β q¯ ϕr + ϕr + ϕr + δrϕ − δ ϕs k l ij N − 1 s r k,l,i,j,r s − permutation k ↔ l

rk r k r k − With the Proposition 3.2.6, by replacing q¯ij with Qi δj +Qj δi 1 r k r k − r k N δi Qj + νδi δj Nνδj δi and expanding, we obtain − q 1,1(P [β,ϕ]D,α) q 1,1(P [α,ϕ]D,β) = 0. − −

So in the case (3.1.19), (δCEq) vanishes and σ not, so the Hoch- schild complex of T V is not formal for V of dimension N > 2.

Suppose now that V is a vectoriel space of any dimension (greater than 2). Choose a subspace U ⊂ V of ﬁnite dimension N ∈ N. If the Hochschild complex associated with the algebra T V was formal, we would have a L -morphism between (g,d) and (G,b + D), ∞ and so d3 = 0. The cohomology class of σ would vanish, and with a good choice of the map q, we would have σ(v1,v2,v3) = 0 for ∈ ⊕ T T + all vi gV = K1 Hom(V, V )/ V , i = 1,2,3. In particular, for ∈ ⊕ T T + ⊂ all u1,u2,u3 gU = K1 Hom(U, U)/ U gV , we would have

σ gU (u1,u2,u3) = 0 which is not the case because the Hochschild complex| of T U is not formal. Thus, for any vector space V , the Hochschild complex associated with the algebra T V is not formal. 64 Chapter 3. Study of the formality for the free algebras 4 Study of the formality for the Lie algebra so(3)

Contents 4.1 Description of the Lie algebra so(3) ...... 66 4.2 Subalgebra of the Chevalley-Eilenberg complex . 67 4.3 Deformation retract of the complex ...... 70 4.4 Computation of the L structure ...... 71 ∞

e examine the formality for the Lie algebra so(3). We recall W its cohomology, and show that the Chevalley-Eilenberg complex retracts on a smaller algebra, which has the same cohomology. We again expose the formality equations and show that they are not satisﬁed. Computation of a perturbed L structure gives only a 3-arity component and as for the case of∞ free algebras, we obtain a L -morphism whose components in arity greater than 3 are zero. ∞

65 66 Chapter 4. Study of the formality for the Lie algebra so(3)

4.1 Description of the Lie algebra so(3)

We consider the Lie algebra so(3). It admits for matrix representation the set of 3-dimensional antisymmetric matrices,

{ ∈ t − } so(3) = M M3(K), M = M (4.1.1) we can also consider it abstractely, generated by three elements with the following relations

| so(3) =< e1,e2,e3 [ei,ei+1] = ei+2 i = 1,2,3 (mod 3) >, (4.1.2) the unmentioned brackets being zero or deduced by antisymmetry. In that case, generators corresponds to the antisymmetric matrices given by

0 0 0   0 0 1 0 −1 0 − e1 = 0 0 1 e2 =  0 0 0 e3 = 1 0 0 0 1 0 −1 0 0 0 0 0

and the bracket to the matrix commutator : [x,y] = xy − yx. We identify so(3) to its dual via the scalar product q, with

1 q(e ,e ) B δ i j 2 ij on the basis and (abusively using the same name)

1 1 q(x) = x.x = (x2 + x2 + x2) 2 2 1 2 3 the associated quadratic form. The Poisson structure associated to the Lie algebra g = so(3) is

1 X P = P ij∂ ∧ ∂ 2 i j 16i,j63 1 = P 12∂ ∧ ∂ + P 13∂ ∧ ∂ + P 21∂ ∧ ∂ + P 23∂ ∧ ∂ + P 31∂ ∧ ∂ + P 32∂ ∧ ∂ 2 1 2 1 3 2 1 2 3 3 1 3 2 1 = x ∂ ∧ ∂ − x ∂ ∧ ∂ − x ∂ ∧ ∂ + x ∂ ∧ ∂ + x ∂ ∧ ∂ − x ∂ ∧ ∂ 2 3 1 2 2 1 3 3 2 1 1 2 3 2 3 1 1 3 2 ∧ ∧ ∧ = x1∂2 ∂3 + x2∂3 ∂1 + x3∂1 ∂2,

so (S(so(3)),·,{ , }) is a Poisson algebra, with

{ } ∧ ∧ ∧ P B , B x1∂2 ∂3 + x2∂3 ∂1 + x3∂1 ∂2 (4.1.3) 4.2. Subalgebra of the Chevalley-Eilenberg complex 67 4.2 Cohomology and subalgebra of the Chevalley- Eilenberg complex

We work with the Lie algebra g B so(3) and its Chevalley-Eilenberg A g S g L Vn g S g complex B CCE( , ) = n N Hom( , ). Since so(3) is simple, using∈ Whitehead’s theorem, its cohomology a with values in S g is given by  [q]1 if k = 0, K k k S a B HCE (g, g) = {0} if k = 1 or k = 2, (4.2.1)  K[q]ω if k = 3, ∧ ∧ where ω = ∂1 ∂2 ∂3. Moreover, the Schouten bracket [ , ]s of the graded Lie algebra A[1] being of degree 0, it induces on the cohomology a[1] a zero ≡ bracket [ , ]s0 0 ; a[1] is thus an abelian Lie algebra.

a[1] = K[q]1 ⊕ {0} ⊕ {0} ⊕ K[q]ω Degree −1 0 1 2 ∈ − Indeed, if f ,g K[q], then [f ,g]s0 = 0 (degree 2), [f ω,gω]s0 = 0 1 { } (degree 4) and [f ,gω]s0 = 0 because a[1] = 0 .

More generally, we compute the Schouten brackets between these four components at the level of the complex A[1].

Proposition 4.2.1 The elements

unit1 ∈ S g, unit for the graded commutative multiplication ∧,

Euler ﬁeld E B x1∂1 + x2∂2 + x3∂3 “trace” derivation, ∧ ∧ ∧ Poisson structure P B x1∂2 ∂3 + x2∂3 ∂1 + x3∂1 ∂2, ∧ ∧ basis trivector ω B ∂1 ∂2 ∂3

generate (on K[q]) a subalgebra Ared[1] =< 1,E,P ,ω > of the graded Lie algebra (A[1],[ , ]s). Lemma 4.2.2 Let f = α(q) ∈ K[q]. We have the following identities.

(i) ∂if = ∂iα(q) = xiα0(q) (iv) [P,E]s = P , for 1 6 i 6 3, (v) [ω,E]s = 3ω, (ii) [E,α(q)1]s = 2qα0(q), ∧ (iii) [P ,α(q)1]s = δCE(α(q)1) = 0, (vi) P E = 2qω.

Proof. 68 Chapter 4. Study of the formality for the Lie algebra so(3)

(i) ∂α ∂q ∂ 1 2 2 2 ∂if = ∂iα(q) = = α0(q) x1 + x2 + x3 = xiα0(q). ∂q ∂xi ∂xi 2

(ii) 3 3 X X E(α(q)) = xi∂i(α(q)) = xixiα0(q) = 2qα0(q). i=1 i=1 (iii) We have

∧ − − 1 ( 1) ∧ [x1∂2 ∂3,α(q)1]s = ( 1) × − [α(q)1,x1∂2 ∂3]s ∧ − 1 0 ∧ = [α(q)1,x1∂2]s ∂3 + ( 1)− × x1∂2 [α(q)1,∂3]s − ∧ − ∧ = x1∂2(α(q)) ∂3 x1∂2 ∂3(α(q)) − − = x1α0(q)(x2∂3 x3∂2) and similarly ∧ − − [x2∂3 ∂1,α(q)]s = x2α0(q)(x3∂1 x1∂3), ∧ − − [x3∂1 ∂2,α(q)]s = x3α0(q)(x1∂2 x2∂1),

thus − − − − [P ,α(q)1]s = α0(q)(x1x2∂3 x1x3∂2+x2x3∂1 x2x1∂3+x3x1∂2 x3x2∂1) = 0.

(iv) − [P,E]s = [E,P ]s − ∧ ∧ ∧ = [x1∂1 + x2∂2 + x3∂3,x1∂2 ∂3 + x2∂3 ∂1 + x3∂1 ∂2]s.

For 1 6 i 6 3, ∧ ∧ − 0 1 ∧ [xi∂i,x1∂2 ∂3]s = [xi∂i,x1∂2]s ∂3 + ( 1) × x1∂2 [xi∂i,∂3]s − ∧ − ∧ = (xiδi1∂2 x1δ2i∂i) ∂3 x1∂2 δ3i∂i and similarly ∧ − ∧ − ∧ [xi∂i,x2∂3 ∂1]s = (xiδi2∂3 x2δ3i∂i) ∂1 x2∂3 δ1i∂i, ∧ − ∧ − ∧ [xi∂i,x3∂1 ∂2]s = (xiδi3∂1 x3δ1i∂i) ∂2 x3∂1 δ2i∂i.

Thus, ∧ − ∧ − ∧ [E,P ]s = x1∂2 ∂3 x2∂3 ∂1 x3∂1 ∂2 − ∧ ∧ − ∧ x1∂2 ∂3 + x2∂3 ∂1 x3∂1 ∂2 − ∧ − ∧ ∧ x1∂2 ∂3 x2∂3 ∂1 + x3∂1 ∂2 = − 2P + P = −P,

and so [P,E]s = P . 4.2. Subalgebra of the Chevalley-Eilenberg complex 69

(v) − ∧ ∧ [ω,E]s = [x1∂1 + x2∂2 + x3∂3,∂1 ∂2 ∂3]s − ∧ ∧ − ∧ ∧ − ∧ ∧ = [E,∂1]s ∂2 ∂3 ∂1 [E,∂2]s ∂3 ∂1 ∂2 [E,∂3]s = 3ω.

(vi) ∧ ∧ ∧ ∧ ∧ P E = (x1∂2 ∂3 + x2∂3 ∂1 + x3∂1 ∂2) (x1∂1 + x2∂2 + x3∂3) 2 ∧ ∧ 2 ∧ ∧ 2 ∧ ∧ = x1∂2 ∂3 ∂1 + x2∂3 ∂1 ∂2 + x3∂1 ∂2 ∂3 = 2qω.

Proof of the Proposition. We have to show the stability of the re- ∈ duced algebra by the Schouten bracket [ , ]s. Let α,β K[q], using the identities of the preceding lemma, we obtain

[α 1,β 1]s = 0,

[αE,β 1]s = αE(β)1 = 2qαβ0 1,

[αP ,β 1]s = α[P ,β 1]s = αδCE(β 1) = 0, ∧ ∧ [αω,β 1]s = α[∂1 ∂2 ∂3,β 1]s ∧ ∧ − ∧ ∧ ∧ ∧ = α [∂1,β 1] ∂2 ∂3 ∂1 [∂2,β 1] ∂3 + ∂1 ∂2 [∂3,β 1] ∧ − ∧ ∧ = α β0x1∂2 ∂3 β0x2∂1 ∂3 + β0x3∂1 ∂2 = αβ0P, − − [αE,βE]s = αE(β)E βE(α)E = 2q(αβ0 α0β)E, − [αP ,βE]s = β[αP ,E]s = β[E,αP ]s − − = βE(α)P βα[E,P ]s = β(α − 2qα0)P, ∧ [αω,βE]s = α[ω,βE]s + [α 1,βE]s ω = α [ω,β 1] ∧ E + β[ω,E] − 2qβα0ω = α β0P ∧ E + 3βω − 2qα0βω = α 2qβ0ω + 3βω − 2qα0βω = 2q(αβ0 − α0β) + 3αβ ω, and [αP ,βP ]s = 0, [αω,βP ]s = 0 [αω,βω]s = 0, the last two equalities for degree reasons. Thus, the various brackets give multiples (in K[q]) of the elements E,P ,ω. 70 Chapter 4. Study of the formality for the Lie algebra so(3)

This reduced algebra can also be written

⊕ ⊕ ⊕ ⊕ Ared[1] = K[q]1 K[q]E K[q]P K[q]ω = H W, (4.2.2) where H B K[q]1⊕K[q]ω = a[1] is the cohomology and W = K[q]E⊕ K[q]ω its supplement in Ared[1].

4.3 Deformation retract of the complex

We will compute the cohomology of the subcomplex (Ared,δCE), we obtain the same cohomology than for the entire Chevalley- S Eilenberg complex A = CCE(g, g).

Theorem 4.3.1 The cohomology of the subcomplex (Ared,δCE) is given by  [q]1 if k = 0, K k { } ared B 0 if k = 1 or k = 2, (4.3.1)  K[q]ω if k = 3.

Proof. Let α ∈ K, we use the Lemma 4.2.2 to obtain the following results.

0 0 0 δCE(α 1) = 0, so Z Ared = Ared = K[q] and ared = K[q] since there are no coboundaries in degree 0.

δCE(αE) = [P ,αE]s = αP so αE is a cocycle if and only if α = 0, 1 { } 1 { } therefore Z Ared = 0 and it follows that ared = 0 . 2 2 δCE(αP ) = 0, so Z Ared = Ared. Moreover, αP is a coboundary, 2 2 2 because αP = δCE(αE), so B Ared = Z Ared and thus ared = {0}. 3 3 3 { } δCE(αω) = [P ,αω]s = 0, so Z Ared = Ared and B Ared = 0 cause ∈ 3 δ(γP ) = 0 for all γ K[q], so ared = K[q]ω.

Thus, the Chevalley-Eilenberg complex A retracts by deformation on the reduced complex Ared, which has the same cohomol- A → A pr A → A ogy. Note i : red and Ared : red the canonical inclusion and projection.

Theorem 4.3.2 The cochain complex (Ared,δCE) is a deformation retract of (A,δCE) i.e. − pr there is a homotopy h of degree 1 between idA et Ared ,

i | / (Ared,δCE A ) o o (A,δCE) h , (4.3.2) red pr g Ared 4.4. Computation of the L structure 71 ∞

− pr map satisfying δh + hδ = idA Ared . k → k 1 Proof. Noting hk : A A − , the map h deﬁned by  h0 = 0   h = 0 h = 1 h (αP ) = αE  2  h3 = 0

ﬁts, moreover, it veriﬁes hδCEh = h.

4.4 Computation of the L∞ structure

We will study the formality equations (2.3.8), working on A[2]. ≡ For the Lie algebra g = so(3), we have D = D2,[ , ]s0 0 so d2 = 0, and a section going back from the cohomology to the cocycles is given by ϕ1 = inc the canonical inclusion of the components of the cohomology in the Chevalley-Eilenberg complex, we will omit it ∈ in the computations. For y1,...,yk+1 a[2], we obtain at the ﬁrst levels the following equations.

Level k = 00= δCE inc, Level k = 1 0 = δCEϕ2(y1,y2) + D(y1,y2), (4.4.1)

Level k = 2

d3(y1,y2,y3) = δCEϕ3(y1,y2,y3) + D(y1,ϕ2(y2,y3)) − y2 y3 + ( 1)| || |D(y2,ϕ2(y1,y3)) (4.4.2) − y3 ( y1 + y2 ) + ( 1)| | | | | | D(y3,ϕ2(y1,y2))

∈ Let α,β K[q]. Since ϕ2 is of degree 0, it is sufficient to consider at the first order k = 1 elements y1 = α 1 and y2 = βω. Writ- ∈ ing ϕ2(α 1,βω) = γE, with γ K[q], we have δCEϕ2(α 1,βω) = [P ,γE]s = γP so ⇒ δCEϕ2(α 1,βω) + D(α 1,βω) = 0 γP = α0βP . → Thus we can define the symmetric graded map ϕ2 : a[2] A[2] by

ϕ2(α 1,β 1) = 0 ϕ2(α 1,βω) = α0βE ϕ2(αω,βω) = 0 72 Chapter 4. Study of the formality for the Lie algebra so(3)

n → ∈ More generally, ϕn : a[2]⊗ A[2] being of degree 0, for i1,...,in {− } i1 in ⊂ i1+ +in 2,1 , ϕn(a[2] ,...,a[2] ) A[2] ··· so, choosing p elements of − − degree 2 and n p elements of degree 1, the image of ϕn is in the component of degree 1(n − p) − 2p = n − 3p. Restricting to the − − ⇔ − subalgebra Ared[2], we have 2 6 n 3p 6 1 n 1 6 3p 6 n + 2. At the level k = 2, the map ϕ3 takes as argument p = 1 (2 6 − 3p 6 5) element of degree 2, noted y1 = α 1 and the other ele- ∈ ments of degrees 1, y2 = β2ω,y3 = β3ω, with α,β2,β3 K[q]. The right-hand side of equation (4.4.2) then writes, using identities of Lemma 4.2.2

δCE(ϕ3(α 1,β2ω,β3ω)) − 1 − 1 ( 2) − 1 − 1 ( 2+1) + ( 1) ( 1) × − [β2ω,ϕ2(α 1,β3ω)]s + ( 1) ( 1) × − [β3ω,ϕ2(α 1,β2ω)]s − = δCE(ϕ3(α 1,β2ω,β3ω)) [β2ω,α0β3E]s + [β3ω,α0β2E]s − = δCE(ϕ3(α 1,β2ω,β3ω)) + 4qα0 β20 β3 β2β30 ω

and the ω multiple is not a coboundary. This shows that there is no formality in this case.

Theorem 4.4.1 The Chevalley-Eilenberg complex of the Lie algebra so(3) is not formal. Proof. In the right-hand side of equation (4.4.2), the multiple (not zero) of ω cannot be written δCE(αP ) because [P ,αP ]s = 0. Thus, we cannot choose a map ϕ3 which would cancel the ω term. If we choose in (4.4.1) another map ϕ20 = ϕ2 + χ2, then δCE(χ2) = 0. ∈ 1 S { } Since χ2 Z (g, g) = 0 , it is a coboundary χ2 = δCE(f ), with ∈ 0 S S f CCE (g, g) = g. But then

− yi D(yi,χ2(yj,yk)) = ( 1)| |[yi,[P ,f (yj,yk)]s]s − = [P,[yi,f (yj,yk)]s]s − yi +1 = ( 1)| | δCED(yi,f (yj,yk))

since [P ,yi]s = 0, and this only modifies the term ϕ3 in the equation (4.4.2), so we cannot cancel d3. In order to have a L -morphism, the differential on the cohomology must have higher∞ order components. Order 3 suffice.

Theorem 4.4.2 There is a L structure (a[1],d) induced by d = d3 and a L -morphism ∞ S → S ∞ ◦ induced by ϕ = ϕ1+ϕ2 noted ϕ : (a[2]) (A[2]) such that δCE + D ϕ = ϕ ◦ d. Proof. Using Theorem 2.5.7 on the contraction

i / a[1] o o (A[1],δCE) g h , p 4.4. Computation of the L structure 73 ∞ we have

◦ X − ◦ r ◦ ◦ ◦ ◦ X − ◦ r ◦ d = ψ1 ( D2 η) D2 ϕ1 = psi1 D2 ( η D2) ϕ1 r N r N ∈ ∈ X − ◦ r ϕ = ( η D2) ϕ1 r N ∈ with ψ1 = p[1] et ϕ1 = i[1].

• • ∈ S Let T3 B α 1 β2ω β3ω (V [1]). Since ϕ1 = id (H[1]), we have S ◦ | ◦X − ◦ n ϕ3 = prV [1] ϕ 3 = prV [1] ( η D2) n N ∈ and • • • − • ϕ3(T3) = prV [1](α 1 β2ω β3 + α0β2E β3ω α0β3E β2ω) = 0, − − ◦ because D2(α 1,βiω) = α0βiP and h(γP ) = γE, the power ( η n • D2) vanish for n > 2 because D2 gives multiples of E ω and η (deﬁned with h) is not zero only on words in P ; ﬁnally the projection on V [1] = S1(V [1]) vanish since the considered elements are words of length greater than 1.

Using that D2(αE,βω) = [βω,αE]s, computation of the term d3 gives again the same expression as before. ◦ ◦ | d3(T3) = ψ1 D2 ϕ 3(T3) − • • − = ψ1( α0β2P β3ω + α0β3P β2ω + D2(α0β2E,β3ω) D2(α0β3E,β2ω)) − = 4qα0(β20 β3 β2β30 )ω

− ◦ n •···• More generally, the iterated compositions ( η D2) on T = α1 1 • •···• ∈ S ⊗S αk 1 β1ω βlω (H[1]) (W [1]) with k > 2 eventually vanish because of the values taken by D2 and the deﬁnition of the chain homotopy h involved in η. Thus, ϕp(T ) = 0 for p > 3 and ◦ ◦ | dp(T ) = ψ1 D2 ϕ p(T ) = 0 for p > 4 with k > 2. 74 Chapter 4. Study of the formality for the Lie algebra so(3) Part II

Hom-algebraic structures Table of Contents

5 Twisting of Hom-(co)algebras 79 5.1 Hom-associative algebras and Hom-Lie algebras . 80 5.2 Hom-coalgebras, Hom-bialgebras and Hom-Hopf algebras ...... 87 5.3 Hom-Lie coalgebras and Hom-Lie bialgebras . . . . 92

6 Hom-(co)Poisson structures 95 6.1 Hom-Poisson algebras ...... 96 6.2 1-operation structures ...... 100 6.3 Hom-coPoisson algebras and duality ...... 103

7 Deformation and quantization of Hom-algebras109 7.1 Formal Hom-deformation ...... 110 7.2 Quantization and twisting of ?-products . . . . . 112 Table of Contents 77 Introduction of the second part

n this second part, we present various Hom-algebraic structures I and ways to deform them. The main characteristic of Hom- algebras and Hom-cogalebras is the fact that classical identities are deformed by an endomorphism. The main part of this work was prepublished in the article [BEM12].

We ﬁrst recall (Chapter 5) the deﬁnitions of Hom-associative and Hom-Lie algebras, as well as the corresponding dual structures ; some examples are given. Duality and twisting are two principles to construct more Hom- structures of the same type than given ones. We explain under which conditions this is possible, and we compute such twist morphisms on some examples.

Hom-Poisson algebras were introduced in [MS10b], where they emerged naturally in the study of 1-parameter formal deformation of commutative Hom-associative algebras. This structure was then studied in [Yau10b]. It is shown that the they are closed under twisting by suitable self maps and under tensor product. Moreover, it is shown that (de)polarized Hom-Poisson algebras are equivalent to admissible Hom-Poisson algebras, each of which only has one binary operation. The twisting principle is also true, we use it to deform the Sklyanin algebra in Example 6.1.5. Section 6.1.4 shows how to build Hom-Poisson algebras starting from Hom-Lie algebras in finite dimension, and introduce such constructions on 3-dimensional examples. Section 6.2 presents the notion of flexible Hom-algebra, more general that Hom-associative algebra, and shows its links with one operation Hom-Poisson algebras. Hopf Hom-coPoisson structures are also defined in Section Section 6.3. Theorem 6.3.5 establish a correspondence between enveloping algebra of Hom- Lie algebra endowed with a Hom-coPoisson structure and Hom- Lie bialgebras. Duality between coPoisson Hopf algebras and Poisson- Hopf algebras is extended to the Hom setting in Theorem 6.3.8.

In Chapter 7, we give the definitions of formal deformation which has been extended for Hom-associatives algebras, Hom-coalgebras and Hom-bialgebras. We finally study the case of the Moyal-Weyl ?-product corresponding to the quantization of the phases space Poisson bracket. Proposition 7.2.6 shows that the twisting morphisms of the (2-dimensional) phases space Poisson bracket are of jacobian 1. To obtain twisting morphisms of the ?-product, we quantize Poisson automorphisms by modifying a differential equation. 78 Table of Contents 5 Twisting usual structures: Hom- algebras and Hom-coalgebras

Contents 5.1 Hom-associative algebras and Hom-Lie algebras . 80 5.1.1 Deﬁnitions ...... 80 5.1.2 Twisting principle ...... 82 5.1.3 Construction of Hom-Lie algebras ...... 83 5.2 Hom-coalgebras, Hom-bialgebras and Hom-Hopf algebras ...... 87 5.2.1 Hom-coalgebras and duality ...... 87 5.2.2 Hom-bialgebra and Hom-Hopf algebra ...... 90 5.3 Hom-Lie coalgebras and Hom-Lie bialgebras .... 92

e brieﬂy present deﬁnitions and properties of usual Hom- W structures; we will show some examples and recall the twisting principle allowing oneself to deform a classical structure into a Hom-structure by means of an algebra morphism. Duality between algebras and coalgebras is also holds for Hom-structures.

79 80 Chapter 5. Twisting of Hom-(co)algebras 5.1 Hom-associative algebras and Hom-Lie algebras

Throughout the next sections, K denotes a ﬁeld of characteristic 0 and A a K-module. In the sequel we denote by σ the linear map 3 → 3 ⊗ ⊗ ⊗ ⊗ σ : A⊗ A⊗ , deﬁned as σ(x1 x2 x3) = x2 x3 x1 and by τij the n → n ⊗···⊗ ⊗···⊗ ⊗···⊗ linear maps τ : A⊗ A⊗ where τij(x1 xi xj xn) = ⊗ ··· ⊗ ⊗ ··· ⊗ ⊗ ··· ⊗ (x1 xj xi xn).

2 We mean by a Hom-algebra a triple (A,µ,α) in which µ : A⊗ → A is a linear map and α : A → A is a linear self map. The linear op 2 → op ◦ map µ : A⊗ A denotes the opposite map, i.e. µ = µ τ12. 2 A Hom-coalgebra is a triple (A,∆,α) in which ∆ : A → A⊗ is a linear map and α : A → A is a linear self map. The linear map op → 2 op ◦ ∆ : A A⊗ denotes the opposite map, i.e. ∆ = τ12 ∆. For a linear self-map α : A → A, we denote by αn the n-fold composition of n copies of α, with α0 = id. A Hom-algebra (A,µ,α) (resp. a 2 Hom-coalgebra (A,∆,α) ) is said to be multiplicative if α◦µ = µ◦α⊗ 2 (resp. α⊗ ◦ ∆ = ∆ ◦ α) . The Hom-algebra is called commutative if µ = µop and the Hom-coalgebra is called cocommutative if ∆ = ∆op. Classical algebras or coalgebras are also regarded as a Hom- algebras or Hom-coalgebras with identity twisting map. Given a Hom-algebra (A,µ,α), we often use the abbreviation µ(x,y) = xy for x,y ∈ A. Likewise, for a Hom-coalgebra (A,∆,α), we will use P ⊗ Sweedler’s notation ∆(x) = (x) x1 x2 but often omit the symbol of summation. When the Hom-algebra (resp. Hom-coalgebra) is multiplicative, we also say that α is multiplicative for µ (resp. ∆).

5.1.1 Deﬁnitions We recall the deﬁnitions of Hom-Lie algebras and Hom-associative algebras.

Deﬁnition 5.1.1 Let (A,µ,α) be a Hom-algebra. 3 → 1. The Hom-associator asµ,α : A⊗ A is deﬁned as

◦ ⊗ − ⊗ asµ,α = µ (µ α α µ). (5.1.1) 2. The Hom-algebra A is called a Hom-associative algebra if it satisﬁes the Hom-associative identity

asµ,α = 0. (5.1.2) 3. A Hom–associative algebra A is called unital if there exists a linear map η : K → A such that 5.1. Hom-associative algebras and Hom-Lie algebras 81

◦ ⊗ ◦ ⊗ µ (idA η) = µ (η idA) = α. (5.1.3)

The unit element is 1A = η(1). 3 → 4. The Hom-Jacobian Jµ,α : A⊗ A is deﬁned as

◦ ⊗ ◦ 2 Jµ,α = µ (α µ) (id + σ + σ ). (5.1.4)

5. The Hom-algebra A is called a Hom-Lie algebra if it satisﬁes µ + µop = 0 and the Hom-Jacobi identity

Jµ,α = 0. (5.1.5)

We recover the usual deﬁnitions of the associator, an associative algebra, the Jacobian, and a Lie algebra when the twisting map α is the identity map. In terms of elements x,y,z ∈ A, the Hom- associator and the Hom-Jacobian are − asµ,α(x,y,z) = (xy)α(z) α(x)(yz),

Jµ,α(x,y,z) = [α(x),[y,z]], x,y,z

where the bracket denotes the product and x,y,z denotes the cyclic sum on x,y,z. Let A = (A,µ,α) and A0 = (A0,µ0,α0) be two Hom-algebras. A linear map f : A → A0 is a morphism of Hom-algebras if

µ0 ◦ (f ⊗ f ) = f ◦ µ and f ◦ α = α0 ◦ f .

It is said to be a weak morphism if holds only the ﬁrst condition.

Proposition 5.1.2 [MS08, Proposition 1.6] To any Hom-associative algebra (A,µ,α), one may associate a Hom-Lie algebra deﬁned for all x,y,z ∈ A by the bracket [x,y] = µ(x,y) − µ(y,x). Proof. The bracket is obviously skewsymmetric and with a direct computation we have

[α(x),[y,z]] + [α(y),[z,x]] + [α(z),[x,y]] = µ(α(x),µ(y,z)) − µ(α(x),µ(z,y)) − µ(µ(y,z),α(x)) + µ(µ(z,y),α(x)) + µ(α(y),µ(z,x)) − µ(α(y),µ(x,z)) − µ(µ(z,x),α(y)) + µ(µ(x,z),α(y)) + µ(α(z),µ(x,y)) − µ(α(z),µ(y,x)) − µ(µ(x,y),α(z)) + µ(µ(y,x),α(z)) = 0. 82 Chapter 5. Twisting of Hom-(co)algebras

{ } Example 5.1.3 Let x1,x2,x3 be a basis of a 3-dimensional vector space A over K. The following multiplication µ and linear map α on A = K3 deﬁne a Hom-associative algebra over K.

µ(x1,x1) = ax1, µ(x2,x2) = ax2, µ(x1,x2) = µ(x2,x1) = ax2, µ(x2,x3) = bx3, µ(x1,x3) = µ(x3,x1) = bx3, µ(x3,x2) = µ(x3,x3) = 0,

α(x1) = ax1, α(x2) = ax2, α(x3) = bx3 where a,b are parameters in K. This algebra is not associative when a , b and b , 0, since − − µ(µ(x1,x1),x3)) µ(x1,µ(x1,x3)) = (a b)bx3. Example 5.1.4 (Jackson sl(2)) The Jackson sl(2) is a q-deformation of the classical Lie algebra sl(2). It carries a Hom-Lie algebra structure but not a Lie algebra structure by using Jackson derivations. It is deﬁned with respect to a basis {e,f ,h} by the brackets and a linear (non multiplicative) map α such that

[h,e] = 2e, α(e) = qe, [h,f ] = −2qf , α(f ) = q2f , q + 1 [e,f ] = h, α(h) = qh, 2 where q is a parameter in K. If q = 1 we recover the classical sl(2).

5.1.2 Twisting principle The following proposition gives an easy way to twist classical structures into Hom-structures.

Theorem 5.1.5 ([Yau09b, Theorem 2.4]) 1. Let A = (A,µ) be an associative algebra and α : A → A be a linear 2 map which is multiplicative with respect to µ, i.e. α◦µ = µ◦α⊗ . ◦ Then Aα = (A,µα = α µ,α) is a Hom-associative algebra. 2. Let A = (A,[ , ]) be a Lie algebra and α : A → A be a linear map which is multiplicative with respect to [ , ], i.e. α ◦ [ , ] = ◦ 2 ◦ [ , ] α⊗ . Then Aα = (A,[ , ]α = α [ , ],α) is a Hom-Lie algebra. Proof. 1. We have as ◦ ⊗ − ⊗ µα,α = µα (µα α α µα) 2 = α ◦ µ ◦ α⊗ ◦ (µ ⊗ id − id ⊗ µ) 2 ◦ = α asµ,id = 0

since (A,µ) is associative, so Aα is a Hom-associative algebra. 5.1. Hom-associative algebras and Hom-Lie algebras 83

2. We have ∀ x,y ∈ A, [y,x] = −[x,y] and ◦ ⊗ ◦ 2 J[ , ]α,α = [ , ]α (α [ , ]α) (id + σ + σ ) 2 2 = α ◦ [ , ] ◦ α⊗ ◦ (id ⊗ [ , ]) ◦ (id + σ + σ ) 2 ◦ = α J[ , ],id = 0

since [ , ] is a Lie bracket, so Aα is a Hom-Lie algebra.

In particular, if α : A → A is an (associative or Lie) algebra morphism (A,µ), then it is also a morphism of the Hom-algebra Aα. Remark 5.1.6 More generally, the categories of Hom-associative algebras and Hom- Lie algebras are closed under twisting by self-weak morphisms. If A = (A,µ,α) is a Hom-associative algebra (resp. Hom-Lie algebra) ◦ ◦ and β a weak morphism, then Aβ = (A,µβ = β µ,β α) is a Hom- associative algebra (resp. Hom-Lie algebra) as well (see [Yau10b]). For an associative (resp. Lie) algebra (A,µ), if α : A → A is a morphism, it is a linear map multiplicative for the structure µ. The twisting principle allows to construct a Hom-associative (resp. ◦ Hom-Lie) algebra (A,µα = α µ,α). If the endomorphism α is an automorphism, the notion of Hom-algebra amounts to the data of an algebra and an automorphism of that algebra. If α is not invertible, the process gives new structures. However, Hom-algebras do not necessarily come from twists of classical structures. The map of Example 5.1.4 of the Hom-Lie Jackson sl(2) algebra is not a morphism for the deﬁned bracket but satisﬁes Hom-Jacobi identity.

5.1.3 Construction of Hom-Lie algebras We expose various methods to construct Hom-Lie algebras. We use the usual Lie algebra sl(2) as a starting point for the diﬀerent cases, but the constructions works more generally. The Lie algebra sl(2) is of dimension 3, generated by elements e,f ,h and relations

[h,e] = 2e, [h,f ] = −2f , [e,f ] = h.

By twisting principle To use the previous twisting principle, we must determinate the morphisms of the initial Lie algebra. For sl(2) which is simple, an endomorphism α : sl(2) → sl(2) has a kernel which is an ideal either reduced to {0} or the whole space sl(2). In the ﬁrst case α is an automorphism, in the second case α ≡ 0. 84 Chapter 5. Twisting of Hom-(co)algebras

Jacobson determinate in [Jac62, Theorem 5,p. 283] the group of automorphisms of sl(2,K): it is the set of mappings sl(2) → sl(2) ∈ 1 with A SL(2,K). X 7→ A− XA The statement is given for a ﬁeld K of characteristic 0 algebraically closed, but the result is again true for K = R. Example 5.1.7 Considering the matrix representation of sl(2) with 0 1 0 0 1 0 e = f = h = , 0 0 1 0 0 −1 a b and a matrix A = ∈ SL(2), we obtain Hom-Lie sl(2) ver- c d α sions of sl(2), with α given by cd d2 α(e) = A 1.e.A = = d2e − c2f + cdh − −c2 −cd −ab −b2 α(f ) = A 1.f .A = = −b2e + a2f − abh − a2 ab bc + ad 2bd α(h) = A 1.h.A = = 2bde − 2acf + (bc + ad)h. − −2ac −bc − ad Specifying parameters, we ﬁnd simpler examples of Hom-Lie 1 versions of sl(2). Taking c = b = 0, we have d = a , and setting λ = d2, [h,e]α = 2λe, α(e) = λe, − 1 1 [h,f ]α = 2λ− f , α(f ) = λ− f , [e,f ]α = h, α(h) = h; − 1 − 2 taking a = d = 0, we have c = b , and setting λ = c

[h,e]α = 2λf , α(e) = λf , − 1 1 [h,f ]α = 2λ− e, α(f ) = λ− e, − − [e,f ]α = h, α(h) = h.

With σ-derivations The original method used in [HLS06] to deform Witt and Virasoro algebras, and which led to the deﬁnition of Hom-Lie algebras, uses operators of σ-derivation. This method gives also deformations of the Lie algebra sl(2) into Hom-Lie algebras. The following is based on the short presentation of [MS10a]. Let A be a commutative associative K-algebra with unit. We denote by σ an endomorphism of A.A σ-derivation on A is a K- → linear map ∂σ : A A such that the σ-twisted Leibniz rule holds: 5.1. Hom-associative algebras and Hom-Lie algebras 85

∂σ (ab) = ∂σ (a)b + σ(a)∂σ (b). (5.1.6)

We let Derσ (A) denote the A-module of σ-derivations of A. Fix- → ∈ ing a homomorphism σ : A A, a σ-derivation ∂σ Derσ (A) and an element δ ∈ A, we assume that these objects satisfy the two following conditions,

⊆ σ(Ann(∂σ )) Ann(∂σ ), (5.1.7) ∀ ∈ ∂σ (σ(a)) = δσ(∂σ (a)), a A, (5.1.8) { ∈ · } · { · ∈ } where Ann(∂σ ) = a A,a ∂σ = 0 . Let A ∂σ = a ∂σ ,a A denote the cyclic A-submodule of Derσ (A) generated by ∂σ and extend σ · · · to A ∂σ by σ(a ∂σ ) = σ(a) ∂σ . The following theorem, from [HLS06], introduces an K-algebra · structure on A ∂σ making it a quasi-Hom-Lie algebra. ⊆ Theorem 5.1.8 ([HLS06, Theorem 3 p.11]) If σ(Ann(∂σ )) Ann(∂σ ), then the bilinear ∈ map [ , ]σ deﬁned for a,b A by

· · · ◦ · − · ◦ · [a ∂σ ,b ∂σ ]σ B (σ(a) ∂σ ) (b ∂σ ) (σ(b) ∂σ ) (a ∂σ ) (5.1.9) · is a well-deﬁned K-algebra product on the K-linear space A ∂σ . It satisﬁes the following identities for a,b,c ∈ A,

· · − · [a ∂σ ,b ∂σ ]σ = (σ(a)∂σ (b) σ(b)∂σ (a)) ∂σ , (5.1.10) · · − · · [a ∂σ ,b ∂σ ]σ = [b ∂σ ,a ∂σ ]σ , (5.1.11)

and if, in addition, (5.1.8) holds, we have the following six-term Jacobi identity · · · · · · · [σ(a) ∂σ ,[b ∂σ ,c ∂σ ]σ ]σ + δ [a ∂σ ,[b ∂σ ,c ∂σ ]σ ]σ = 0. (5.1.12) a,b,c We apply this on the representation of sl(2) in terms of ﬁrst order diﬀerential operators acting on some vector space A of functions in the variable t.

e 7→ ∂, h 7→ −2t∂, f 7→ −t2∂. ∈ → Example 5.1.9 Fix q K∗. Let A = K[t] and σ : A A be the unique endomor- ∈ phisme defined by σ(t) B qt, i.e. σ(f (t)) = f (qt) for f A, and ∂σ : → n { } n A A the σ-derivation defined by ∂σ (t) = t, i.e. ∂σ (t ) = n qt , { } ··· n 1 with q-analogues n q = 1 + q + + q − . · Set S the K-linear subspace of A ∂σ spanned by elements E B − − 2 ∂σ , H B 2t∂σ , F B t ∂σ . The element δ = q satisfies equation 86 Chapter 5. Twisting of Hom-(co)algebras

(5.1.8), so with the previous bracket [ , ]σ and the K-linear map α = σ + qid,(S,[ , ]σ ,α) is a Hom-Lie algebra. We have

[H,E]σ = 2E, α(E) = (1 + q)E, − [H,F]σ = 2qF, α(F) = q(1 + q)F, q + 1 [E,F] = H, α(H) = 2qH. σ 2 For q = 1, brackets are those of sl(2) but α = 2id.

Example 5.1.10 Slightly adapting the previous relations, we obtain a q-deformation 1 family of Witt algebras. We consider this time A = C[t,t− ], and, id − σ with the previous notations, D B t∂ = − . Then σ q − 1

D M · − n · q B Derσ (A) = C dn where dn = t D. n Z ∈ D → D With the bracket [ , ]: q q deﬁned on generators by [dn,dm] B { } − { } n q m q dn+m and the linear map α = σ + id, deﬁned on ele- n D ments by α(dn) = (q + 1)dn,( q,[ , ],α) is a Hom-Lie algebra. For q = 1, we recover the classical Witt algebra.

Hom-Lie structure associated with a bracket An other way to construct Hom-Lie algebras is to compute the various possible linear maps (not necessarily multiplicatives) for a given bracket. We apply this idea to the previous exemple.. q + 1 Example 5.1.11 Considering brackets [h,e] = 2e, [h,f ] = −2qf , [e,f ] = h, we 2 compute maps α written as α = (αij) satisfying Hom-Jacobi identity. We solve equations x,y,z[α(x),[y,z]] = 0 on the basis, in the coeﬃcients αij. We obtain Hom-Lie algebras with the previous brackets and map α such that

α(e) = ae + bf + ch, 1 + q α(f ) = de + aqf + k h, 4 4q α(h) = ke + c f + lh, 1 + q

with parameters a,b,c,d,k,l ∈ K. Details of the computation are given in annex Section A.2. Setting b = c = d = k = 0, a = 1 + q, l = 2q, we obtain Exam- ple 5.1.9, and for b = c = d = k = 0, a = l = q, we get Exam- ple 5.1.4. These maps are never multiplicative for the brackets of the usual Lie algebra sl(2), so these Hom-Lie algebras do not 5.2. Hom-coalgebras, Hom-bialgebras and Hom-Hopf algebras 87

come from the classical Lie algebra sl(2) by twisting principle The- orem 5.1.5. Also, these maps are not multiplicatives for the considered bracket. In particular, for q = 1, Hom-Lie algebras with the same bracket that sl(2) are obtained with the following linear maps α (mentioned in [MS10b]),

α(e) = ae + bf + ch, 1 α(f ) = de + af + kh, 2 α(h) = ke + 2cf + lh.

5.2 Hom-coalgebras, Hom-bialgebras and Hom- Hopf algebras

In this section we summarize and describe some of basic properties of Hom-coalgebras, Hom-bialgebras and Hom-Hopf algebras which generalize the classical coalgebra, bialgebra and Hopf algebra structures.

5.2.1 Hom-coalgebras and duality

Deﬁnition 5.2.1 A Hom-coalgebra is a triple (A,∆,α) where A is a K-module and ∆ : A → A ⊗ A, α : A → A are linear maps. A Hom-coassociative coalgebra is a Hom-coalgebra satisfying

(α ⊗ ∆) ◦ ∆ = (∆ ⊗ α) ◦ ∆. (5.2.1) A Hom-coassociative coalgebra is said to be counital if there exists a map ε : A → K satisfying

(id ⊗ ε) ◦ ∆ = α and (ε ⊗ id) ◦ ∆ = α. (5.2.2)

Let (A,∆,α) and (A0,∆0,α0) be two Hom-coalgebras (resp. Hom- coassociative coalgebras). A linear map f : A → A0 is a morphism of Hom-coalgebras (resp. Hom-coassociative coalgebras) if

(f ⊗ f ) ◦ ∆ = ∆0 ◦ f and f ◦ α = α0 ◦ f .

If furthermore the Hom-coassociative coalgebras admit counits ε and ε0, we have moreover ε = ε0 ◦ f . The following theorem shows how to construct a new Hom- coassociative Hom-coalgebra starting with a Hom-coassociative Hom- coalgebra and a Hom-coalgebra morphism. We only need the coassociative comultiplication of the coalgebra. 88 Chapter 5. Twisting of Hom-(co)algebras

Theorem 5.2.2 Let (A,∆,α,ε) be a counital Hom-coalgebra and β : A → A be a weak ◦ ◦ Hom-coalgebra morphism. Then (A,∆β,α β,ε), where ∆β = ∆ β, is a counital Hom-coassociative coalgebra. ◦ Proof. We show that (A,∆β,α β) satisﬁes the axiom (5.2.1). Indeed, using the fact that (β ⊗ β) ◦ ∆ = ∆ ◦ β, we have ◦ ⊗ ◦ ◦ ⊗ ◦ ◦ ◦ α β ∆β ∆β = (α β ∆ β) ∆ β = ((α ⊗ ∆) ◦ ∆) ◦ β2 = ((∆ ⊗ α) ◦ ∆) ◦ β2 ⊗ ◦ ◦ = ∆β α β ∆β.

Moreover, the axiom (5.2.2) is also satisﬁed, since we have ⊗ ◦ ⊗ ◦ ◦ ◦ ⊗ ◦ ◦ ⊗ ◦ (id ε) ∆β = (id ε) ∆ β = α β = (ε id) ∆ β = (ε id) ∆β.

Remark 5.2.3 The previous theorem shows that the category of coassociative Hom- coalgebras is closed under weak Hom-coalgebra morphisms. It leads to the following examples:

1. Let (A,∆) be a coassociative coalgebra and β : A → A be a coalgebra morphism. Then (A,∆β,β) is a Hom-coassociative coalgebra.

2. Let (A,∆,α) be a multiplicative coassociative Hom-coalgebra. n+1 For all non negative integer n,(A,∆αn ,α ) is a Hom-coassociative coalgebra.

In the following we show that there is a duality between Hom- associative and Hom-coassociative structures (see [SPAS09, MS10a]).

Theorem 5.2.4 Let (A,∆,α) be a coalgebra Hom-coassociative. Then the dual (A∗,∆∗,α∗) is a Hom-associative algebra. Moreover A∗ is unital if A is counital.

Proof. The product µ = ∆∗ is deﬁned from A∗ ⊗ A∗ to A∗ by

h ⊗ i ⊗ X ∀ ∈ (f g)(x) = ∆∗(f ,g)(x) = ∆(x),f g = (f g)(∆(x)) = f (x1)g(x2), x A (x)

where h·,·i is the natural pairing between the vector space A ⊗ A and its dual vector space. For f ,g,h ∈ A∗ and x ∈ A, we have

(f g)α∗(h)(x) = h(∆ ⊗ α) ◦ ∆(x),f ⊗ g ⊗ hi and

α∗(f )(gh)(x) = h(α ⊗ ∆) ◦ ∆(x),f ⊗ g ⊗ hi 5.2. Hom-coalgebras, Hom-bialgebras and Hom-Hopf algebras 89

so the Hom-associativity µ ◦ (µ ⊗ α∗ − α∗ ⊗ µ) = 0 comes from the Hom-coassociativity (∆ ⊗ α − α ⊗ ∆) ◦ ∆ = 0. Moreover, if A has a counit ε satisfying (id ⊗ε)◦∆ = α = (ε ⊗id)◦∆ then for f ∈ A∗ and x ∈ A we have X X (εf )(x) = ε(x1)f (x2) = f (ε(x1)x2) = f (α(x)) = α∗(f )(x) (x) (x) and X X (f ε)(x) = f (x1)ε(x2) = f (x1ε(x2)) = f (α(x)) = α∗(f )(x) (x) (x)

which shows that ε is the unit in A∗. The dual of a Hom-algebra (A,µ,α) is not always a Hom-coalgebra, because the coproduct does not land in the good space: µ∗ : A∗ → ⊗ ⊗ (A A)∗ ) A∗ A∗. It is the case if the Hom-algebra is of ﬁnite dimension, since (A ⊗ A)∗ = A∗ ⊗ A∗. In the general case, for any algebra A, deﬁne

A◦ = {f ∈ A∗,f (I) = 0 for some cofinite ideal I of A}, where a cofinite ideal I is an ideal I ⊂ A such that A/I is finite dimensional. A◦ is a subspace of A∗ since it is closed under multiplication by scalars and the sum of two elements of A◦ is again in A◦ since the intersection of two cofinite ideals is again such. If A is finite dimensional, of course A◦ = A∗.

Lemma 5.2.5 Let (A,µA,αA) and (B,µB,αB) be two Hom-associative algebras and f : A → B a morphism of Hom-algebras. Then the dual map f ∗ : B∗ → A∗ satisﬁes f ∗(B◦) ⊂ A◦. Proof. Let J be a coﬁnite ideal of B and p : B → B/J the canonical projection. We set fe= p ◦ f : A → B/J. 1 Remark that f − (J) is an ideal of A. Indeed, for x ∈ A we have 1 1 1 1 f (xf − (J)) = f (x)f (f − (J)) = f (x)J ⊂ J. So xf − (J) ⊂ f − (J). More- 1 1 ⊂ 1 over αA(f − (J)) = f − (αB(J)) f − (J). The sequence

1 i fe 0 → f − (J) −→ A −→ B/J → 0 1 → 1 is exact. Deﬁne a map f? : A/f − (J) B/J by f?(x+f − (J)) = f (x). It 1 1 induces an isomorphism A/f − (J) → Imfe. Thus A/f − (J) is ﬁnite- dimensional. Similarly, we have f ∗(B◦) ⊂ A◦. Indeed, let b∗ ∈ B∗ such that 1 Ker(b∗) ⊃ J. Then Ker(f ∗(b∗)) ⊃ f − (J), because 1 1 hf ∗(b∗),f − (J)i = hb∗,f (f − (J))i = hb∗,Ji = 0. 90 Chapter 5. Twisting of Hom-(co)algebras

Using this lemma, we can show in a similar way as [Swe69, Lemme 6.0.1] that A◦ ⊗ A◦ = (A ⊗ A)◦ and the dual µ∗ : A∗ → (A ⊗ A)∗ of the multiplication µ : A ⊗ A → A satisfies µ∗(A◦) ⊂ A◦ ⊗ A◦. Indeed, for f ∈ A∗, x,y ∈ A, we have hµ∗(f ),x ⊗ yi = hf ,xyi. Hence, if I is a cofinite ideal such that f (I) = 0, then I⊗A+A⊗I is a cofinite ideal of A ⊗ A on which µ∗(f ) vanishes. For a map f : A → B we note f f | : B → A . Set ∆ µ = ◦ B ∗ B◦ ◦ ◦ B ◦ µ | and ε : A → defined by ε(f ) = f (1). ∗ A◦ ◦ K

Theorem 5.2.6 Let (A,µ,α) be a multiplicative Hom-associative algebra. Then (A◦,∆,α◦) is an Hom-coalgebra. Moreover, it is counital if A is unital.

Proof. The coproduct ∆ is deﬁned from A◦ to A◦ ⊗ A◦ by

∆(f )(x ⊗ y) = µ | (f )(x ⊗ y) = hµ(x ⊗ y),f i = f (xy), x,y ∈ A. ∗ A◦

For f ,g,h ∈ A◦ and x,y ∈ A, we have

(∆ ◦ α◦) ◦ ∆(f )(x ⊗ y ⊗ z) = hµ ◦ (µ ⊗ α)(x ⊗ y ⊗ z),f i and

(α◦ ◦ ∆) ◦ ∆(f )(x ⊗ y ⊗ z) = hµ ◦ (⊗µα)(x ⊗ y ⊗ z),f i

so the Hom-coassociativity (∆⊗α◦ −α◦ ⊗∆)◦∆ = 0 comes from the Hom-associativity µ ◦ (µ ⊗ α − α ⊗ µ) = 0. Moreover, if A has a unit η satisfying µ ◦ (id ⊗ η) = α = µ ◦ (η ⊗ id) then for f ∈ A◦ and x ∈ A we have

(ε ⊗ id) ◦ ∆(f )(x) = f (1.x) = f (α(x)) = α◦(f )(x) and

(id ⊗ ε) ◦ ∆(f )(x) = f (x.1) = f (α(x)) = α◦(f )(x)

→ 7→ which shows that ε : A◦ K, f f (1) is the counit in A◦.

If the Hom-associative algebra is ﬁnite-dimensional, its dual is a Hom-coassociative coalgebra.

5.2.2 Hom-bialgebra and Hom-Hopf algebra

Deﬁnition 5.2.7 A Hom-bialgebra is a tuple (A,µ,α,η,∆,β,ε) where

1. (A,µ,α,η) is a Hom-associative algebra with a unit η.

2. (A,∆,β,ε) is a Hom-coassociative coalgebra with a counit ε. 5.2. Hom-coalgebras, Hom-bialgebras and Hom-Hopf algebras 91

3. The linear maps ∆ and ε are compatible with the multiplication µ and the unit η, that is

∆(e) = e ⊗ e where e = η (1), ⊗ • X ⊗ ⊗ ⊗ ∆(µ(x y)) = ∆(x) ∆(y) = µ(x1 y1) µ(x2 y2), (x)(y) ε (e) = 1, ε (µ(x ⊗ y)) = ε (x)ε (y), ε ◦ α (x) = ε (x),

where the bullet • denotes the multiplication on tensor product. If α = β the Hom-bialgebra is noted (A,µ,η,∆,ε,α). Combining previous observations, with only one twisting map, we obtain:

Proposition 5.2.8 Let (A,µ,∆,α) be a Hom-bialgebra. Then the ﬁnite dual (A◦,µ◦,∆◦,α◦) is a Hom-bialgebra as well.

Remark 5.2.9

1. Given a Hom-bialgebra (A,µ,α,η,∆,β,ε), it is shown in [SPAS09, MS10a] that the vector space Hom(A,A) with the multiplication given by the convolution product carries a structure of Hom-associative algebra.

2. An endomorphism S of A is said to be an antipode if it is the inverse of the identity over A for the Hom-associative algebra Hom(A,A) with the multiplication given by the convolution product deﬁned by

f ? g = µ ◦ (f ⊗ g) ◦ ∆

and the unit being η ◦ ε.

3.A Hom-Hopf algebra is a Hom-bialgebra with an antipode.

By combining Theorem 5.1.5 and Theorem 5.2.2, we obtain:

Proposition 5.2.10 Let (A,µ,∆,α) be a Hom-bialgebra and β : A → A be a Hom-bialgebra ◦ morphism commuting with α. Then (A,µβ,∆β,β α) is a Hom-bialgebra. Notice that the category of Hom-bialgebra is not closed under weak Hom-bialgebra morphisms.

Example 5.2.11 (Universal enveloping Hom-algebra) Given a Hom-associative algebra A = (A,µ,α), one can associate to it a Hom-Lie algebra HLie(A) = (A,[ , ],α) with the same underlying module (A,α) and the bracket 92 Chapter 5. Twisting of Hom-(co)algebras

[ , ] = µ − µop. This construction gives a functor HLie from Hom- associative algebras to Hom-Lie algebras. In [Yau08], Yau constructed the left adjoint UHLie of HLie. He also made some minor modiﬁcations in [Yau10a] to take into account the unital case. The functor UHLie is deﬁned as

M M n UHLie(A) = FHNAs(A)/I∞ with FHNAs(A) = Aτ⊗ (5.2.3) wt n N τ Tn ∈ ∈ wt for a Hom-Lie algebra (A,[ , ],α). Here Tn is the set of weighted n- trees encoding the multiplication of elements (by trees) and twist- n n ing by α (by weights), Aτ⊗ is a copy of A⊗ and I∞ is a certain submodule of relations build in such a way that the quotient is Hom-associative. → ⊗ Moreover, the comultiplication ∆ : UHLie(A) UHLie(A) UHLie(A) deﬁned on A by ∆(x) = α(x)⊗1+1⊗α(x) equips the Hom-associative algebra UHLie(A) with a structure of Hom-bialgebra.

5.3 Hom-Lie coalgebras and Hom-Lie bialgebras

As it is the case for the Hom-associative algebras, the Hom-Lie algebras also have a dualized version, Hom-Lie coalgebras. They share the same kind of properties. We review here the principal results, similar results can be found in [Yau09a].

Deﬁnition 5.3.1 A Hom-Lie coalgebra (A,∆,α) is a Hom-coalgebra satisfying ∆ + ∆op = 0 and the Hom-coJacobi identity

(id + σ + σ 2) ◦ (α ⊗ ∆) ◦ ∆ = 0. (5.3.1) We call ∆ the cobracket.

We recover a Lie coalgebra when α = id. Just like (co)associative (co)algebras we have the following dualization properties.

Theorem 5.3.2

1. Let (A,∆,α) be a Hom-Lie coalgebra. Then (A∗,∆∗,α∗) is an Hom-Lie algebra.

2. Let (A,[ , ],α) be a Hom-Lie algebra. Then (A◦,[ , ]◦,α◦) is an Hom-Lie coalgebra.

The twisting principle also works, showing that the category of Hom-Lie coalgebras is closed under weak Hom-coalgebras morphisms. 5.3. Hom-Lie coalgebras and Hom-Lie bialgebras 93

Theorem 5.3.3 Let (A,∆,α) be a Hom-Lie coalgebra and β : A → A a weak Hom- ◦ ◦ coalgebra morphism. Then (A,∆β = ∆ β,α β) is a Hom-Lie coalgebra. op op ◦ Proof. We have ∆β + ∆β = (∆ + ∆ ) β = 0, and 2 ◦ ◦ ⊗ ◦ 2 ◦ ◦ ⊗ ◦ ◦ ◦ (id + σ + σ ) (α β ∆β) ∆β = (id + σ + σ ) (α β ∆ β) ∆ β = (id + σ + σ 2) ◦ (α ⊗ ∆) ◦ ∆ ◦ β2 = 0.

The previous theorem can be used to construct Hom-Lie coalgebras.

Corollary 5.3.4 1. Let (A,∆) be a Lie coalgebra and β : A → A be a Lie coalgebra morphism. Then (A,∆β,β) is a Hom-Lie coalgebra. 2. Let (A,∆,α) be a multiplicative Hom-Lie coalgebra. For all non n+1 negative integer n, (A,∆αn ,α ) is a Hom-Lie coalgebra. The Hom-Lie bialgebra structure was introduced first in [Yau09a]. The definition presented below is slightly more general. They bor- der the class defined by Yau and permit to consider the compatibility condition for different A-valued cohomology of Hom-Lie algebras.

Deﬁnition 5.3.5 A Hom-Lie bialgebra is a tuple (A,[ , ],α,∆,β) where 1.( A,[ , ],α) is a Hom-Lie algebra. 2.( A,∆,β) is a Hom-Lie coalgebra. 3. The following compatibility condition holds for x,y ∈ A:

− ∆([x,y]) = adα(x)(∆(y)) adα(y)(∆(x)), (5.3.2) n → n where the adjoint map adx : A⊗ A⊗ (n > 2) is given by

n ⊗ ··· ⊗ X ⊗ ··· ⊗ ⊗ ⊗ ⊗ ··· ⊗ adx(y1 yn) = β(y1) β(yi 1) [x,yi] β(yi+1) β(yn). (5.3.3) − i=1

A morphism f : A → A0 of Hom-Lie bialgebras is a linear map com- 2 2 muting with α and β such that f ◦[ , ] = [ , ]◦f ⊗ and ∆◦f = f ⊗ ◦∆. If α = β = id we recover Lie bialgebras and if α = β, we recover the class deﬁned in [Yau09a], these Hom-Lie bialgebras are denoted (A,[ , ],∆,α). 94 Chapter 5. Twisting of Hom-(co)algebras

In terms of elements, the compatibility condition (5.3.2) writes

− ∆([x,y]) =adα(x)(∆(y)) adα(y)(∆(x)) ⊗ ⊗ =[α(x),y1] β(y2) + β(y1) [α(x),y2] (5.3.4) − ⊗ − ⊗ [α(y),x1] β(x2) β(x1) [α(y),x2].

Remark 5.3.6 If α = β, the compatibility condition (5.3.4) is equivalent to say that ∆ is a 1-cocycle with respect to α0-adjoint cohomology of Hom-Lie 1 algebras and for β = id, it corresponds to α− -adjoint cohomology, (see [She11] and [AEM11]). The following Proposition generalizes slightly [Yau09a, Theorem 3.5].

Proposition 5.3.7 Let (A,[ , ],∆,α) be a Hom-Lie bialgebra and β : A → A a Hom-Lie ◦ bialgebra morphism commuting with α. Then (A,[ , ]β = β [ , ],∆β = ∆ ◦ β,α ◦ β) is a Hom-Lie bialgebra. ◦ Proof. We already know that (A,[ , ]β,β α) is a Hom-Lie algebra ◦ and that (A,∆β,α β) is a Hom-Lie coalgebra. It remains to prove the compatibility condition (5.3.4) for ∆β and [ , ]β, with the twisting map α ◦ β = β ◦ α. On one side, we have

◦ 2 ◦ 22 ◦ ∆β([x,y]) = ∆ β [x,y] = β⊗ ∆([x,y]),

2 2 since ∆ ◦ β = β⊗ ◦ ∆. Using in addition β ◦ [ , ] = [ , ] ◦ β⊗ and the fact that α and β commute, we have ⊗ adα β(x)(∆β(y)) = adα β(x)(β(y1) β(y2)) ◦ ◦ ◦ ⊗ ◦ 2 ◦ 2 ⊗ ◦ = [α β(x),β(y1)]β α β (y2) + α β (y1) [α β(x),β(y2)]β 22 ⊗ ⊗ = β⊗ ([α(x),y1] α(y2) + α(y1) [α(x),y2]). − It follows that ∆β([x,y]) = adα β(x)(∆β(y)) adα β(y)(∆β(x)) as wished. ◦ ◦

As for the Hom-bialgebra, the category of Hom-Lie bialgebra is not closed under weak Hom-Lie bialgebra morphisms. Hom-Lie bialgebra can be dualized. We obtain the following proposition which generalizes the result stated in [Yau09a] for ﬁ- nite dimensional case using natural pairing.

Proposition 5.3.8 Let (A,[ , ],α,∆,β) be a multiplicative Hom-Lie bialgebra. Then the ﬁnite dual (A◦,[ , ]◦,α◦,∆◦,β◦) is a Hom-Lie bialgebra as well. 6 Hom-(co)Poisson structures

Contents 6.1 Hom-Poisson algebras ...... 96 6.1.1 Deﬁnitions and examples ...... 96 6.1.2 Twisting principle ...... 97 6.1.3 Application to Sklyanin algebra ...... 97 6.1.4 Constructing Hom-Poisson algebras from Hom- Lie algebras ...... 98 6.2 1-operation structures ...... 100 6.2.1 Flexibles Hom-algebras ...... 100 6.2.2 Link with Hom-Poisson algebras ...... 101 6.3 Hom-coPoisson algebras and duality ...... 103 6.3.1 Link with Hom-Lie bialgebras ...... 103 6.3.2 Duality ...... 105

om-Poisson algebras also satisfy the twisting principle, such a H morphism is computed for the Sklyanin algebra. We show the links between Hom-Poisson algebras and 1-operation structures, then study their duals, which are linked to Hom-Lie bialgebras.

95 96 Chapter 6. Hom-(co)Poisson structures 6.1 Hom-Poisson algebras

We give here the deﬁnition of Hom-Poisson algebras and show that the twisting principle also holds in that case. We present in ﬁnite dimension a way to build Hom-Poisson algebras from Hom-Lie algebras.

6.1.1 Deﬁnitions and examples Deﬁnition 6.1.1 A Hom-Poisson algebra is tuple (A,µ,{ , },α) consisting of

(1) a commutative Hom-associative algebra (A,µ,α) and

(2) a Hom-Lie algebra (A,{ , },α) such that the Hom-Leibniz identity

{ } ◦ ⊗ ◦ ⊗ { } { } ⊗ ◦ , (µ α) = µ (α , + ( , α) τ23) (6.1.1) is satisﬁed. In a Hom-Poisson algebra (A,{ , },µ,α), the operation { , } is called Hom-Poisson bracket. In terms of elements x,y,z ∈ A, the Hom-Leibniz identity says

{xy,α(z)} = α(x){y,z} + {x,z}α(y)

where as usual µ(x,y) is abbreviated to xy. By the antisymmetry of the Hom-Poisson bracket { , }, the Hom-Leibniz identity is equivalent to {α(x),yz} = {x,y}α(z) + α(y){x,z}. We recover Poisson algebras when the twisting map is the identity.

Deﬁnition 6.1.2 A Hom-Poisson bialgebra (A,µ,η,∆,ε,α,{ , }) is a Hom-Poisson algebra (A,µ,{ , },α) which is also a Hom-bialgebra (A,µ,η,∆,ε,α), the two structures being compatible in the sense that { , } is a coderivation along µ,

∆ ◦ { , } = ({ , } ⊗ µ + µ ⊗ { , }) ◦ ∆[2].

In term of elements, this compatibility condition writes

∆({a,b}) = {∆(a),∆(b)}

with the Hom-Poisson bracket on A ⊗ A deﬁned by { ⊗ ⊗ } { } ⊗ ⊗ { } a1 a2,b1 b2 B a1,b1 a2b2 + a1b1 a2,b2 .

We have the same deﬁnition for Hom-Poisson Hopf algebras. 6.1. Hom-Poisson algebras 97

{ } Example 6.1.3 Let x1,x2,x3 be a basis of a 3-dimensional vector space A over K. The following multiplication µ, skew-symmetric bracket { , } and linear map α on A deﬁne a Hom-Poisson algebra over K3: { } µ(x1,x1) = x1, x1,x2 = ax2 + bx3, { } µ(x1,x2) = µ(x2,x1) = x3, x1,x3 = cx2 + dx3,

α(x1) = λ1x2 + λ2x3, α(x2) = λ3x2 + λ4x3, α(x3) = λ5x2 + λ6x3

where a,b,c,d,λ1,λ2,λ3,λ4,λ5,λ6 are parameters in K.

6.1.2 Twisting principle Theorem 6.1.4 ([Yau10b, Theorem 3.2]) Let A = (A,µ,{ , }) be a Poisson algebra, and α : A → A be a linear map which is multiplicative for µ and { , }. Then ◦ { } ◦{ } Aα = (A, µα = α µ, , α = α , , α) is a Hom-Poisson algebra.

Proof. Using Theorem 5.1.5, we already have that (A,µα,α) is a { } commutative Hom-associative algebra and that (A, , α,α) is a Hom-Lie algebra. It remains to check the Hom-Leibniz identity { } ◦ ⊗ ◦ { } ◦ 2 ◦ ⊗ , α (µα α) = α , α⊗ (µ id) = α2 ◦ { , } ◦ (µ ⊗ id), since A is a Poisson algebra 2 ◦ ◦ ⊗ { } { } ⊗ ◦ = α µ (id , + ( , id) τ23) ◦ ◦ 2 ◦ ⊗ { } { } ⊗ ◦ = α µ α⊗ (id , + ( , id) τ23) ◦ ⊗ { } { } ⊗ ◦ = µα (α , α + ( , α α) τ23),

so Aα is a Hom-Poisson algebra.

6.1.3 Application to Sklyanin algebra E Example 6.1.5 The Sklyanin Poisson algebra q4( ) (see [ORTP10] for a more detailed definition and properties) is defined on C[x0,x1,x2,x3] by a parameter k ∈ C with the usual polynomial multiplication, and bracket given by { , } with brackets between the coordinate functions defined as { } 2 − xi,xi+1 = k xixi+1 xi+2xi+3, i = 0,1,2,3 (mod 4). { } 2 − 2 xi,xi+2 = k(xi+3 xi+1), E → E We again search a morphism α : q4( ) q4( ) written as α = (αij) with respect to the basis (x0,x1,x2,x3), by solving the coefficients αij in the equations α([xi,xj]) = [α(xi),α(xj)] with respect to the basis. For simplicity, we take α diagonal, αij = 0 if i , j. E E We obtain q4( )α, Hom-Poisson versions of q4( ), with α given by 98 Chapter 6. Hom-(co)Poisson structures

− − 1. α(x0) = λx0, α(x1) = iλx1, α(x2) = λx2, α(x3) = iλx3, − − 2. α(x0) = λx0, α(x1) = iλx1, α(x2) = λx2, α(x3) = iλx3, − − 3. α(x0) = λx0, α(x1) = λx1, α(x2) = λx2, α(x3) = λx3, 4. α = λid, with λ ∈ K. E For example, q4( ) carries a structure of Hom-Poisson algebra, for any λ ∈ C, with the following bracket { } − 2 2 − xi,xi+1 = λ (k xixi+1 xi+2xi+3), i = 1,2,3,4 (mod 4), { } 2 2 − 2 xi,xi+2 = λ k(xi+3 xi+1), and linear map − − α(x0) = λx0, α(x1) = λx1, α(x2) = λx2, α(x3) = λx3.

6.1.4 Constructing Hom-Poisson algebras from Hom-Lie algebras Suppose that (A,[ , ],α) is a finite-dimensional Hom-Lie algebra { } k and ei 16i6n is a basis of A. Let Cij be the structure constants of Pn k the bracket with respect to the basis, that is [ei,ej] = k=1 Cijek and s Pn s αi be the coefficients of the morphism α, that is α(ei) = s=1 αi es. The skew-symmetry of the bracket and the Hom-Jacobi condition can be written with the structure constants as k − k Cji = Cij antisymmetry, X p q p q p q s (αi Cjk + αj Cki + αk Cij)Cpq = 0 Hom-Jacobi identity. 16p,q6n To construct a Hom-Poisson algebra from a Hom-Lie algebra, we should define a commutative multiplication · which is Hom- associative and a bracket { , } satisfying the Hom-Leibniz identity. Define the bracket { , } as being equal to the bracket [ , ] on the basis, and extended by the Hom-Leibniz identity. k · Let Mij be the structure constants for the multiplication, that is ei Pn k k k ej = k=1 Mijek. By commutativity, Mji = Mij. The Hom-Leibniz identity writes { · } − ·{ } − { }· 0 = ei ej,α(ek) α(ei) ej,ek ei,ek α(ej) n ⇔ X p q s − p q p q s 0 = Mijαk Cpq (αi Cjk + Cikαj )Mpq es s=1 | {z } Sijks ⇔ 0 = Sijks,

6.1. Hom-Poisson algebras 99

l 4 3 1 giving a linear system in the Mij of n equations in n unknowns . The Hom-associativity writes

· · − · · 0 = (ei ej) α(ek) α(ei) (ej ek) n ⇔ X p q − p q s 0 = (Mijαk αi Mjk)Mpq es s=1 | {z } Rijks ⇔ 0 = Rijks,

l 4 3 giving a non linear system in the Mij of n equations in n unknowns. Solving ﬁrst the equations of Hom-Leibniz and then checking if the solutions satisfy the Hom-associativity equations, we obtain example of Hom-Poisson algebras. { } Example 6.1.6 We consider the 3-dimensional Hom-Lie algebra with basis e1,e2,e3 , brackets given by

2 3 [e1,e2] = C12e2 + C12e3 2 3 [e1,e3] = C13e2 + C13e3 [e2,e3] = 0,

0 0 0 { } and morphism α = 0 b 0 in the basis e1,e2,e3 . The only multi- 0 0 b plications giving a Hom-Poisson algebra are of the form

· e1 e2 = λe2 · e1 e3 = λe3 · e2 e3 = 0 · ei ei = 0 for i = 1,2,3.

Example 6.1.7 Other examples of Hom-Poisson algebras of dimension 3 with basis { } e1,e2,e3 are given by twisting the following Poisson algebra:

· { } e1 e1 = e2 e1,e3 = ae2 + be3,

with all other multiplication and brackets equal to zero. The morphism α is computed to be multiplicative for the multiplication and the bracket.

1 n2(n+1) actually 2 unknowns using the commutativity 100 Chapter 6. Hom-(co)Poisson structures

With a , 0, b , 0 With a , 0, b = 0

α(e1) = α21e2 α(e2) = 0 α(e3) = α13e1 + α23e2 + α33e3

α(e1) = α11e1 + α21e2 + α31e3 α(e1) = α11e1 + α21e2 + α31e3 2 2 α(e2) = α11e2 α(e2) = α11e2 − 2 −a 2 α11α33 α11 α(e3) = α11e2 α(e3) = e1 + α23e2 + α33e3 b α31

α(e1) = α11e1 + α21e2 2 α(e2) = α11e2 α(e3) = α13e1 + α23e2 + α11e3

6.2 1-operation structures

We recall here some results on ﬂexible structures described in [MS08] and provide a connection with Hom-Poisson algebras. Al- gebras with one operation were introduced by Loday and studied by Markl and Remm in [MR06]. The twisted version was studied in [Yau10b] where they are called admissible Hom-Poisson algebras.

6.2.1 Flexibles Hom-algebras Deﬁnition 6.2.1 A Hom-algebra A = (A,µ,α) is called ﬂexible if for any x,y ∈ A

µ(µ(x,y),α(x)) = µ(α(x),µ(y,x))). (6.2.1)

Remark 6.2.2 Using the Hom-associator asµ,α, the condition (6.2.1) may be written as asµ,α(x,y,x) = 0. Lemma 6.2.3 Let A = (A,µ,α) be a Hom-algebra. The following assertions are equivalent

1. A is flexible. ∈ 2. For any x,y A, asµ,α(x,y,x) = 0. ∈ − 3. For any x,y,z A, asµ,α(x,y,z) = asµ,α(z,y,x). Proof. The equivalence of the first two assertions follows from the − − definition. The assertion asµ,α(x z,y,x z) = 0 holds by definition and it is equivalent to asµ,α(x,y,z) + asµ,α(z,y,x) = 0 by linearity. 6.2. 1-operation structures 101

Corollary 6.2.4 Any Hom-associative algebra is ﬂexible. Let A = (A,µ,α) be a Hom-algebra, where µ is the multiplication and α a twisting map. We deﬁne two new multiplications using µ:

∀ x,y ∈ A x • y = µ(x,y) + µ(y,x), {x,y} = µ(x,y) − µ(y,x).

+ We set A = (A,•,α) and A− = (A,{ , },α). Proposition 6.2.5 A Hom-algebra A = (A,µ,α) is ﬂexible if and only if

{α(x),y • z} = {x,y}• α(z) + α(y) •{x,z}. (6.2.2) Proof. Let A be a ﬂexible Hom-algebra. By Lemma (6.2.3), this is ∈ equivalent to asµ,α(x,y,z) + asµ,α(z,y,x) = 0 for any x,y,z A. This implies

asµ,α(x,y,z) + asµ,α(z,y,x) + asµ,α(x,z,y) + (6.2.3) − − asµ,α(y,z,x) asµ,α(y,x,z) asµ,α(z,x,y) = 0

By expansion, the previous relation is equivalent to {α(x),y • z} = {x,y}• α(z) + α(y) •{x,z}. Conversely, assume we have the condition (6.2.2) in Proposition. By setting x = z in (6.2.3), one gets asµ,α(x,y,x) = 0. Therefore A is ﬂexible.

6.2.2 Link with Hom-Poisson algebras Deﬁnition 6.2.6 A 1-operation Hom-Poisson algebra is a Hom-algebra (A,·,α) satisfying, for any x,y,z ∈ A,

· · · · − · · − · · 3as ,α(x,y,z) = (x z) α(y) + (y z) α(x) (y x) α(z) (z x) α(y). (6.2.4) · If α is the identity map, A is called a 1-operation Poisson algebra. We consider a Hom-algebra (A,·,α). We deﬁne two operations • : A ⊗ A → A and { , } : A ⊗ A → A by

∀ x,y ∈ A, x • y = x · y + y · x, {x,y} = x · y − y · x. (6.2.5)

Theorem 6.2.7 (A,•,{ , },α) is a Hom-Poisson algebra if and only if (A,·,α) is a 1- operation Hom-Poisson algebra. Proof. Suppose that (A,•,{ , },α) is a Hom-Poisson algebra. Since

1 ∀ x,y ∈ A, x · y = ({x,y} + x • y), 2 102 Chapter 6. Hom-(co)Poisson structures

we have, by expansion,

· · − · · 1{ { }} as ,α(x,y,z) = (x y) α(z) α(x) (y z) = α(y), z,x , · 4 and, on the other hand, using that the multiplication • is Hom- associative and commutative, and that { , } is a Hom-Lie bracket, 3 (x · z) · α(y) + (y · z) · α(x) − (y · x) · α(z) − (z · x) · α(y) = {α(y),{z,x}}. 4 We thus have the equation (6.2.4). Suppose now that the equation (6.2.4) is veriﬁed. We have to show that • is Hom-associative and that { , } is a Hom-Lie bracket. Using the relation (6.2.4), we obtain the identities

∀ ∈ x,y,z A as ,α(x,y,z) + as ,α(y,z,x) + as ,α(z,x,y) = 0 (6.2.6) ∀ ∈ · · · x,y,z A as ,α(x,y,z) + as ,α(z,y,x) = 0. (6.2.7) · · This last identity (6.2.7) shows that (A,·,α) is a Hom-ﬂexible algebra using Lemma 6.2.3. We now obtain • • − • • as ,α(x,y,z) = (x y) α(z) α(x) (y z) • − =as ,α(y,z,x) + as ,α(x,z,y) (as ,α(z,y,x) + as ,α(x,y,z)) · · · · (6.2.7) = 0.

So the product • is Hom-associative and commutative by deﬁni- tion. Moreover, { { }} { { }} { { }} J , ,α(x,y,z) = α(x), y,z + α(y), z,x + α(z), x,y −{ } = (as ,α(x,y,z) + as ,α(y,z,x) + as ,α(z,x,y))+ · · · as ,α(x,z,y) + as ,α(y,x,z) + as ,α(z,y,x) · · · (6.2.6) = 0,

so { , } is a Hom-Lie bracket. Since A is ﬂexible, Proposition 6.2.5 leads to the compatibility between • and { , },

{α(x),y • z} = {x,y}• α(z) + α(y) •{x,z}.

So (A,•,{ , },α) is a Hom-Poisson algebra.

Proposition 6.2.8 Let (A,·) be a 1-operation Poisson algebra, and α : A → A be a linear 2 map multiplicative for the multiplication ·, i.e. α ◦ · = · ◦ α⊗ , then · ◦ · Aα = (A, α = α ,α) is a 1-operation Hom-Poisson algebra. Proof. We have 6.3. Hom-coPoisson algebras and duality 103

· · − · · 3as ,α(x,y,z) = (x α y) α α(z) α(x) α (y α z) ·α = α(α(x · y) · α(z)) − α(α(x) · α(y · z)) = α2((x · y) · z − x · (y · z)),

and since · veriﬁes the 1-operation equation,

2 · · · · − · · − · · 3as ,α(x,y,z) = α ((x z) y + (y z) x (y x) z (z x) y) ·α = α(α(x · z) · α(y)) + α(α(y · z) · α(x)) − α(α(y · x) · α(z)) − α(α(z · x) · α(y)) · · · · − · · − · · = (x α z) α (y) + (y α z) α α(x) (y α x) α α(z) (z α x) α α(y).

6.3 Hom-coPoisson algebras and duality

In this section, we extend the connection between Lie bialgebras and coPoisson-Hopf algebras presented in [CP94] to the Hom setting. We also extend to Hom-algebras the result stated in [OP11], that the Hopf dual of a coPoisson Hopf algebra is a Poisson-Hopf algebra.

6.3.1 Link with Hom-Lie bialgebras

Deﬁnition 6.3.1 A Hom-coPoisson algebra consists of a cocommutative coassociative Hom-coalgebra (A,∆,ε,α) equipped with a skew-symmetric linear map δ : A → A ⊗ A, the Hom-coPoisson cobracket, satisfying the following conditions

(i) (Hom-coJacobi identity)

(id + σ + σ 2) ◦ (α ⊗ δ) ◦ δ = 0, (6.3.1)

(ii) (Hom-coLeibniz rule)

⊗ ◦ ⊗ ◦ ◦ ⊗ ◦ (∆ α) δ = (α δ) ∆ + τ23 (δ α) ∆. (6.3.2)

It is denoted by a tuple (A,∆,ε,α,δ).

Proposition 6.3.2 Let (A,∆,ε,α,δ) be a Hom-coPoisson algebra and β : A → A be a Hom- ◦ ◦ ◦ coPoisson algebra morphism. Then (A,∆β = ∆ β,ε,α β,δβ = δ β) is a Hom-coPoisson algebra. 104 Chapter 6. Hom-(co)Poisson structures

◦ Proof. Theorem 5.2.2 insures that (A,∆β,ε,α β) is a coassociative ◦ Hom-coalgebra and Theorem 5.3.3 that (A,δβ,α β) is a Hom-Lie coalgebra. It remains to show the compatibility condition (6.3.2). On the left-hand side, we have ⊗ ◦ ◦ ◦ ⊗ ◦ ◦ ◦ ⊗ ◦ 2 (∆β α β) δβ = (∆ β α β) δ β = (∆ α) β ,

and the right-hand side gives ◦ ⊗ ◦ ◦ ⊗ ◦ ◦ (α β δβ) ∆β + τ23 (δβ α β) ∆β ◦ ⊗ ◦ ◦ ◦ ◦ ◦ ⊗ ◦ ◦ ◦ = (α β δ β) ∆ β + τ23 (δ β α β) ∆ β ⊗ ◦ ◦ ⊗ ◦ ◦ 2 = [(α δ) ∆ + τ23 (δ α) ∆] β ,

which ends the proof. We may state the following Corollaries. Starting from a classical coPoisson algebra, we may construct Hom-coPoisson algebras using coPoisson algebra morphisms. On the other hand a Hom- coPoisson algebra gives rise to inﬁnitely many Hom-coPoisson algebras.

Corollary 6.3.3

1. Let (A,∆,ε,δ) be a coPoisson algebra and β : A → A be a coPois- ◦ ◦ son algebra morphism. Then (A,∆β = ∆ β,ε,β,δβ = δ β) is a Hom-coPoisson algebra.

2. Let (A,∆,ε,α,δ) be a Hom-coPoisson algebra. Then for any non negative integer n, we have (A,∆ ◦ αn,ε,αn+1,δ ◦ αn) is a Hom- coPoisson algebra.

Deﬁnition 6.3.4 A Hom-coPoisson bialgebra (A,µ,η,∆,ε,α,δ) is a Hom-coPoisson algebra (A,∆,ε,α,δ) which is also a Hom-bialgebra (A,µ,η,∆,ε,α), the two structures being compatible in the sense that δ is a derivation along ∆, δ ◦ µ = µ[2] ◦ (δ ⊗ ∆ + ∆ ⊗ δ). A Hom-coPoisson Hopf algebra (A,µ,η,∆,ε,S,α,δ) is a Hom-coPoisson bialgebra (A,µ,η,∆,ε,α,δ) with an antipode S, such that the tuple (A,µ,η,∆,ε,S,α) is a Hom-Hopf algebra.

Theorem 6.3.5 Let (g,[ , ],α) be a Hom-Lie algebra. If its universal enveloping algebra ◦ 2 ◦ UHLie(g) has a Hom-coPoisson structure δ such that δ α = α⊗ δ, making it a Hom-coPoisson bialgebra, then δ(g) ⊂ g⊗g, and δ|g equips (g,[ , ],α,δ|g,id) with a structure of Hom-Lie bialgebra. Conversely, for a Hom-Lie bialgebra (g,[ , ],δ,α), the cobracket δ : g → g ⊗ g extends uniquely to a Hom-coPoisson cobracket on UHLie(g), which makes UHLie(g) a Hom-coPoisson bialgebra. 6.3. Hom-coPoisson algebras and duality 105

→ ⊗ Proof. Let δ : UHLie(g) UHLie(g) UHLie(g) be a Hom-coPoisson ⊂ ⊗ ∈ cobracket on UHLie(g). To show that δ(g) g g, let x g, and P (1) ⊗ (2) (1) (2) ∈ write δ(x) = (x) x x where x ,x UHLie(g). We may assume that the x(2) are linearly independent. By the Hom-coLeibniz condition (6.3.2), we have X ∆(x(1)) ⊗ α(x(2)) =α(1) ⊗ δ(α(x)) + α(α(x)) ⊗ δ(1) (x) ◦ ⊗ ⊗ + τ23 (δ(1) α(α(x)) + δ(α(x)) α(1)) since x ∈ g and ∆(x) = 1⊗α(x) + α(x) ⊗ 1. Moreover, α(1) = 1 and δ is a derivation along ∆ so δ(1) = 0, hence

X (1) ⊗ (2) ⊗ ◦ ⊗ ∆(x ) α(x ) = 1 δ(α(x)) + τ23 (δ(α(x)) 1) (x) X = 1⊗α(x(1)) + α(x(1)) ⊗ 1 ⊗ α(x(2)) (x)

2 (1) using the multiplicativity δ ◦ α = α⊗ ◦ δ. It follows that the x ⊗ ⊗ are Hom-primitive elements (∆(x) = 1 α(x)+α(x) 1) of UHLie(g), ⊂ ⊗ hence δ(g) g UHLie(g). Since δ is skew-symmetric, ⊂ ⊗ ∩ ⊗ ⊗ δ(g) (g UHLie(g)) (UHLie(g) g) = g g.

To prove the compatibility condition (5.3.4) for δ|g and the twisting maps α and id, let x,y ∈ g and compute

δ([x,y]) =δ(xy − yx) =δ(x)∆(y) + ∆(x)δ(y) − δ(y)∆(x) − ∆(y)δ(x) =[∆(x),δ(y)] − [∆(y),δ(x)] =[α(x),y(1)] ⊗ y(2) + y(1) ⊗ [α(x),y(2)] − [α(y),x(1)] ⊗ x(2) − x(1) ⊗ [α(y),x(2)] − =adα(x)(∆(y)) adα(y)(∆(y)). → ⊗ → ⊗ Conversely, δ : g g g uniquely extends in δ : UHLie(g) UHLie(g) | UHLie(g) such that δ g = δ, with the formula

δ(xy) = δ(x)∆(y) + ∆(x)δ(y).

This gives UHLie(g) a structure of Hom-coPoisson bialgebra.

6.3.2 Duality Deﬁnition 6.3.6 An algebra A over K is said to be an almost normalizing extension over K if A is a ﬁnitely generated K-algebra with generators x1,...,xn satisfying the condition 106 Chapter 6. Hom-(co)Poisson structures

n − ∈ X xixj xjxi Kxl + K (6.3.3) l=1 for all i,j.

Lemma 6.3.7 Let A be an almost normalizing extension of K with generators x1,...,xn. Then A is spanned by all standard monomials

r1 r2 ··· rn ∈ x1 x2 xn , ri N together with the unity 1. Proof. This follows immediately from induction on the degree of monomials.

Recall that ﬁnite dual A◦ of a Hom-bialgebra A consists of

A◦ = {f ∈ A∗,f (I) = 0 for some coﬁnite ideal I of A},

where A∗ is the dual vector space of A. Theorem 6.3.8 Let A be a Hom-coPoisson bialgebra with Poisson co-bracket δ and twisting map α. If A is an almost normalizing extension over K, then the ﬁnite dual A◦ is a Hom-Poisson bialgebra with twisting map α◦ and bracket

{f ,g}(x) = hδ(x),f ⊗ gi, x ∈ A (6.3.4)

for any f ,g ∈ A◦, where h·,·i is the natural pairing between the vector space A ⊗ A and its dual vector space. Proof. The proof is almost the same as in [OP11]. We do not re- produce here the ﬁrst step showing that the bracket (6.3.4) is well- deﬁned, it uses the fact that A is an almost normalizing extension over K. ◦ − The skew-symmetry follows from τ12 δ = δ, we have { } h ⊗ i h ◦ ⊗ i g,f (x) = δ(x),g f = τ12 δ,f g = −hδ(x),f ⊗ gi = −{f ,g}(x),

for all x ∈ A. The equation (6.3.4) satisﬁes the Hom-Leibniz rule: since

{f g,α◦(h)}(x) = h(∆ ⊗ α) ◦ δ(x),f ⊗ g ⊗ hi

and

(α◦(f ){g,h} + {f ,h}α◦(g))(x) h ⊗ ◦ ⊗ ⊗ i h ◦ ⊗ ◦ ⊗ ⊗ i = (α δ) ∆(x),f g h + τ23 (δ α) ∆(x),f g h 6.3. Hom-coPoisson algebras and duality 107

for x ∈ A and f ,g,h ∈ A◦, it is enough to show that ⊗ ◦ ⊗ ◦ ◦ ⊗ ◦ (∆ α) δ = (α δ) ∆ + τ23 (δ α) ∆, but this is just the Hom-coLeibniz rule for δ. The equation (6.3.4) satisﬁes the Hom-Jacobi identity: we have

{α◦(f ),{g,h}}(x) = h(α ⊗ δ) ◦ δ(x),f ⊗ g ⊗ hi, {α◦(g),{h,f }}(x) = hσ ◦ (α ⊗ δ) ◦ δ(x),f ⊗ g ⊗ hi, 2 {α◦(h),{f ,g}}(x) = hσ ◦ (α ⊗ δ) ◦ δ(x),f ⊗ g ⊗ hi,

for x ∈ A and f ,g,h ∈ A◦. Hence (6.3.4) satisfies the Hom-Jacobi identity if and only if δ satifies (id + σ + σ 2) ◦ (α ⊗ δ) ◦ δ = 0, which is just the Hom-coJacobi identity of δ. The bracket defined by (6.3.4) satisfies the compatibility condition with the comultiplication of the Hom-bialgebra, it is a µ- coderivation: since δ is a ∆-derivation, we have for f ,g ∈ A◦ ∆({f ,g})(x ⊗ y) = {f ,g}(xy) = hδ(xy),f ⊗ gi = hδ(x)∆(y),f ⊗ gi + h∆(x)δ(y),f ⊗ gi h ⊗ ih ⊗ i = δ(x),f1 g1 ∆(y),f2 g2 h ⊗ ih ⊗ i + ∆(x),f1 g2 δ(y),f2 g2 { } { } = f1,g1 (x)(f2g2)(y) + (f1g1)(x) f2,g2 (y) = {∆(f ),∆(g)}(x ⊗ y).

Finally, the bracket deﬁned by (6.3.4) equips A◦ with the structure of a Hom-Poisson bialgebra, the twisting map being α◦.

Let (g,[ , ],δ,α) be a Hom-Lie bialgebra, UHLie(g) the universal enveloping Hom-bialgebra of g with comultiplication ∆. The cobracket δ : g → g ⊗ g is extended uniquely to a ∆-derivation → ⊗ | δ : UHLie(g) UHLie(g) UHLie(g) such that δ g = δ and δ(xy) = δ(x)∆(y) + ∆(x)δ(y). Then UHLie(g) is a Hom-coPoisson bialgebra with cobracket δ.

Corollary 6.3.9 Let (g,[ , ],δ,α) be a ﬁnite dimensional Hom-Lie bialgebra. Then the dual UHLie(g)◦ of the universal enveloping Hom-bialgebra UHLie(g) is a Hom-Poisson bialgebra with Poisson bracket { } h ⊗ i ∈ f ,g (x) = δ(x),f g , x UHLie(g) ∈ for f ,g UHLie(g)◦. { } Proof. Let x1,...,xn be a basis of g. Then UHLie(g) is an almost normalizing extension over K with generators x1,...,xn. Thus the result follows from Theorem 6.3.8. 108 Chapter 6. Hom-(co)Poisson structures 7 Deformation and quantization of Hom-algebras

Contents 7.1 Formal Hom-deformation ...... 110 7.1.1 Formal deformation of Hom-associative algebras 110 7.1.2 Deformations of Hom-coalgebras and Hom-Bialgebras112 7.2 Quantization and twisting of ?-products ...... 112 7.2.1 Twists of Moyal-Weyl ?-product ...... 114 7.2.2 Twists of the Poisson bracket ...... 116 7.2.3 Quantization of the Poisson automorphisms . . . 118

he theory of formal deformation can be extended to Hom-al- T gebraic structures. We give the deﬁnitions for Hom-algebras, Hom-coalgebras and Hom-bialgebras, as well as some results gen- eralising the classical case. We then seek the Poisson morphisms and morphisms of the ?-product of Moyal-Weyl, to obtain Hom- Poisson and Hom-associative algebras by twist.

109 110 Chapter 7. Deformation and quantization of Hom-algebras 7.1 Formal Hom-deformation

Let A be a K vector space. Let K[[t]] be the power series ring in one variable t and coefficients in K and A[[t]] be the set of formal power series whose coefficients are elements of A,(A[[t]] is obtained by extending the coefficients domain of A from K to K[[t]]). Then A[[t]] is a K[[t]]-module. When A is finite-dimensional, we have A[[t]] = A ⊗ K[[t]]. Note that A is a submodule of A[[t]]. We define formal deformations for Hom-associative algebras, Hom- coassociative coalgebras and Hom-bialgebras.

7.1.1 Formal deformation of Hom-associative algebras

Deﬁnition 7.1.1 Let A = (A,µ0,α0) be a Hom-associative algebra. A formal Hom- associative deformation of A is given by a K[[t]]-bilinear map µt : ⊗ → → A[[t]] A[[t]] A[[t]] and a K[[t]]-linear map αt : A[[t]] A[[t]] of the form

X i X i µt = µit and αt = αit (7.1.1) i>0 i>0 ⊗ → where each µi is a K-bilinear map µi : A A A (extended to → be K[[t]]-bilinear) and each αi is a K-linear map αi : A A (extended to be K[[t]]-linear) such that the following formal Hom- associativity condition holds:

as ◦ ⊗ − ⊗ µt,αt = µt (µt αt αt µt) = 0. (7.1.2)

If αt = id the deﬁnition reduces to formal deformation of an associative algebra already presented in Section 1.2.4.

Example 7.1.2 We can consider the Jackson Hom-Lie algebra sl(2) from Exam- ple 5.1.4 as a formal deformation of sl(2). Setting q = 1 + t, the brackets and twist map are given by

[h,e]t = 2e, αt(e) = e + et, − − [h,f ]t = 2f 2f t, 2 αt(f ) = f + 2f t + f t , h [e,f ] = h + t, α (h) = h + ht, t 2 t The coeﬃcients of the powers of t are the following. − [h,e]0 = 2e, [h,f ]0 = 2f , [e,f ]0 = h, α0(e) = e α0(f ) = f α0(h) = h h [h,e] = 0, [h,f ] = −2f , [e,f ] = , 1 1 1 2 α1(e) = e α1(f ) = 2f α1(h) = h 7.1. Formal Hom-deformation 111

α2(f ) = f

We recover the classical bracket of sl(2) at order 0 and α0 = id. There are other possibles twisting morphisms (given in Example 5.1.11) for the same bracket [ , ]t, for example

2 + t X∞ (−1)kf h α (e) = e, α (f ) = f = f + tk, α (h) = h + t. t t 2(1 + t) 2 t 2 k=0 The equation (7.1.2) can be written

XXX − i+j+k µi(αk(x),µj(y,z)) µi(µj(x,y,αk(z)) t = 0. (7.1.3) i N j N k N ∈ ∈ ∈ Introducing the α-associators 7→ ◦ − (x,y,z) µi α µj(x,y,z) B µi(α(x),µj(y,z)) µi(µj(x,y,α(z)), the deformation equation may be written as follows

s s k XXX ◦ i+j+k X s XX− ◦ (µi α µj)t = 0 or t µi α µs k i = 0. k k − − i N j N k N s N k=0 i=0 ∈ ∈ ∈ ∈ It is equivalent to the inﬁnite system:

s s k XX− ◦ ∈ µi α µs k i = 0, s N. (7.1.4) k − − k=0 i=0 The A-valued Hochschild type cohomology of Hom-associative algebras initiated in [MS10b] and extended in [AEM11] suits and leads to the following cohomological interpretation: 1. There is a natural bijection between H2(A,A) and the set of equivalence classes of deformation (mod t2) of A. 2. If H2(A,A) = 0 then every deformation of A is trivial. The fact that the antisymmetrization of the ﬁrst order element of a deformation of an associative algebra deﬁnes a Poisson bracket remains true in the Hom setting. More precisely, we have the following theorem.

Theorem 7.1.3 ([MS10b, Theorem 4.9]) Let A = (A,µ0,α) be a commutative Hom-associative algebra and At = (A,µt,αt) be a deformation of A. Consider the bracket ∈ { } − deﬁned for x,y A by x,y = µ1(x,y) µ1(y,x) where µ1 is the ﬁrst { } order element of the deformation µt. Then (A,µ0, , ,α0) is a Hom- Poisson algebra. The proof is mainly computational, it leans on the properties of the α-associators, and on rewriting the deformation equations (7.1.4) in terms of coboundary operators. 112 Chapter 7. Deformation and quantization of Hom-algebras

7.1.2 Deformations of Hom-coalgebras and Hom-Bialgebras

Deﬁnition 7.1.4 Let (A,∆,α) be a Hom-coalgebra. A formal Hom-coalgebra defor- → ⊗ mation of A is given by linear maps ∆t : A[[t]] A[[t]] A[[t]] → X i X i and αt : A[[t]] A[[t]] such that ∆t = ∆it and αt = αit i>0 i>0 → ⊗ → where each ∆i, αi are linear maps ∆i : A A A and αi : A A (extended to be K[[t]]-linear) such that the following formal Hom- coassociativity condition holds:

⊗ − ⊗ ◦ (∆t αt αt ∆t) ∆t = 0. (7.1.5) Deﬁnition 7.1.5 Let (A,µ,∆,α) be a Hom-bialgebra. A formal Hom-bialgebra de- ⊗ → formation of A is given by linear maps µt : A[[t]] A[[t]] A[[t]], → ⊗ → ∆t : A[[t]] A[[t]] A[[t]] and αt : A[[t]] A[[t]] such that

1. (A[[t]],µt,αt) is a Hom-associative algebra,

2. (A[[t]],∆t,αt) is a Hom-coassociative coalgebra,

3. the multiplication and the comultiplication are compatible, that is ◦ ⊗ ◦ ◦ ⊗ ∆t µt = (µt µt) τ23 (∆t ∆t).

It is shown in [DM] that deformations are controlled by Hoch- schild type cohomology and any deformation of an unital Hom- associative algebra (resp. counital Hom-coassociative colgebra) is equivalent to an unital Hom-associative algebra (resp. counital Hom-coassociative colgebra). Furthermore, any deformation of a Hom-Hopf algebra as a Hom-bialgebra is automatically a Hom- Hopf algebra. In a similar way as for Hom-associative algebra, we have the following result.

Theorem 7.1.6 Let A = (A,∆0,α) be a cocommutative Hom-coassociative coalgebra and At = (A,∆t,αt) be a deformation of A. Consider the cobracket deﬁned ∈ − op for x A by δ(x) = ∆1(x) ∆1 (x) where ∆1 is the ﬁrst order element of the deformation ∆t. Then (A,∆0,δ,α0) is a Hom-coPoisson algebra.

7.2 Quantization and twisting of ?-products

We set the problem of quantization by deformation for Hom-associative algebras. Let (A,·,{ , },α) be a commutative Hom-associative algebra endowed with a Hom-Poisson bracket { , }. 7.2. Quantization and twisting of ?-products 113

Deﬁnition 7.2.1 A ?-product on A is a one parameter formal deformation deﬁned on A by

X∞ r f ?t g = t µr(f ,g) r=0 such that 1. the ?-product in A[[t]] is Hom-associative, that is r ∀ ∈ X − r N, (µi(µr i(f ,g),α(h)) (µi(α(f ),µr i(g,h))) = 0, − − i=0 · 2. µ0(f ,g) = f g, − { } 3. µ1(f ,g) µ1(g,f ) = f ,g , ∀ 4. µr(f ,1) = µr(1,f ) = 0 r > 0. Remark 7.2.2 1 • The condition 2. shows that [f ,g] B (f ? g − g ? f ) is a t t t deformation of the Hom-Lie structure { , }. − { } • The condition µ1(f ,g) µ1(g,f ) = f ,g expresses the correspondence between the deformation and the Hom-Poisson structure f ? g − g ? f t t | = {f ,g}. t t=0 Similarly we set the dual version of the quantization problem as follows. Let A be cocommutative Hom-coPoisson bialgebra (resp. Hopf algebra) and let δ be its Poisson cobracket. A quantization of A is a Hom-bialgebra (resp. Hom-Hopf algebra) deformation At of A such that op ∆ (a) − ∆ (a) δ(x) = t t (mod t), t where x ∈ A and a is any element of A[[t]] such that x = a (mod t).

Theorem 7.2.3 Let (A,?) be an associative deformation of an associative algebra (A,µ0), X i → with ? = µit . Let α : A A be a morphism such that for all i N ∈ ◦ ∈ ◦ 2 ◦ i N, α µi = µi α⊗ . Then (A,?α = α ?,α) is a Hom-associative deformation of (A,µ0,α,α). ∈ ◦ ◦ 2 ◦ Proof. Since for all i N, α µi = µi α⊗ , we also have α ? = 2 ? ◦ α⊗ . Thus α is an algebra morphism of (A,?) which endows (A,?α,α) of a Hom-associative structure by the twisting principle Theorem 5.1.5. The ﬁrst term of this Hom-associative deformation is µ0,α which is also Hom-associative. 114 Chapter 7. Deformation and quantization of Hom-algebras

7.2.1 Twists of Moyal-Weyl ?-product In the following, we twist the Moyal-Weyl product. It is the associative ?-product corresponding to the deformation of the Poisson phase-space bracket, one of the ﬁrst examples of Kontsevich formal deformation [Kon03]. We consider the Poisson algebra of polynomials of two variables R = (R[x,y],·,{ , }) where the Poisson bracket of two polyno- ∂f ∂g ∂f ∂g mials is given by {f ,g} = − . ∂x ∂y ∂y ∂x The associated associative algebra is (R[x,y][[λ]],?MW ) where the ?-product is given by the Moyal-Weyl formula

λ ◦ 2 (∂x ∂y ∂y ∂x) ?MW = µ0 e ⊗ − ⊗ (7.2.1) which, expanding the exponentials writes on elements f ,g ∈ R[x,y] as

n n X∞ 1 λ X n k n k n k n k k f ? g = (−1) − (∂ ∂ − f )(∂ − ∂ g), (7.2.2) MW n! 2 k x y x y n=0 k=0 noting λ the formal parameter. This ?-product is equivalent to the ?-product which writes more ◦ λ∂x ∂y simply ? = µ0 e ⊗ , on elements, we obtain X ∂nf ∂ng λn X f ? g = = µ (f ,g)λn, (7.2.3) ∂xn ∂yn n! n n N n N ∈ ∈ 1 ∂nf ∂ng where µ (f ,g) = . n n! ∂xn ∂yn The isomorphism between the associative algebras (R[x,y][[λ]],?MW ) and (R[x,y][[λ]],?) is given by

λ 2 ∂x∂y ◦ ◦ ⊗ S = e− S ? = ?MW (S S). → ◦ ◦ 2 Proposition 7.2.4 A morphism α : R[x,y][[λ]] R[x,y][[λ]] satisfying α µi = µi α⊗ ∈ for all i N which gives (R[x,y],?α = α?,α) a structure of Hom- associative algebra is of the form 1 α(x) = ax+b and α(y) = y +c where a,b,c ∈ R, a , 0. (7.2.4) a Proof. Let α : R[x,y] → R[x,y] be a morphism such that for all ∈ 2 i N, αµi = µiα⊗ . In particular, for i = 0, α(f g) = α(f )α(g), (7.2.5)

which shows that α is multiplicative, so it is suﬃcient to deﬁne it on x and y. For i = 1, we get 7.2. Quantization and twisting of ?-products 115

∂f ∂g ∂α(f ) ∂α(g) α = , (7.2.6) ∂x ∂y ∂x ∂y which implies that α({f ,g}) = {α(f ),α(g)}. We set P1(x,y) B α(x) and P2(x,y) B α(y). For f (x,y) = x and g(x,y) = y, the equation (7.2.6) gives ∂P ∂P 1 = α(1) = 1 2 , ∂x ∂y 1 so we must have P (x,y) = ax +Q (y) and P (x,y) = y +Q (x) with 1 1 2 a 2 ∈ \{ } ∈ a R 0 and Q1,Q2 R[x,y]. For f (x,y) = y and g(x,y) = x, the equation (7.2.6) gives

∂P2 ∂P1 0 = α(0) = = Q0 Q0 . ∂x ∂y 2,x 1,y

So we can suppose that Q1(y) = b is constant. Reporting in the equation (7.2.6), with f (x,y) = g(x,y) = y, we ﬁnd

∂P2 ∂P2 1 0 = α(0) = = Q0 , ∂x ∂y 2,x a

so Q2(x) = c is constant. It remains to verify that for i > 1, αµi = i i i i 2 ∈ ∂ f ∂ g 1 ∂ α(f ) ∂ α(g) 1 µiα⊗ i.e. for f ,g R[x,y], α = . By ∂xi ∂yi i! ∂xi ∂yi i! multiplicavity of α, the only non trivial case is f (x,y) = xn and g(x,y) = ym. We have

i i m i ∂ f ∂ g 1 n n i m m i 1 n m n i 1 − 1 α = α i! x − i! y − = i! i! (ax + b) − y + c ∂xi ∂yi i! i i i! i i a i! m i 1 m i i i n ∂ y + c n m n i i 1 − 1 1 ∂ (ax + b) a 1 = i! i! (ax + b) − a y + c = i i a a i! ∂xi ∂yi i! ∂iα(f ) ∂iα(g) 1 = . ∂xi ∂yi i!

The Hom-algebra (R[x,y][[λ]],?α,α) is Hom-associative and not associative if α , id. Indeed, for f (x,y) = 1, g(x,y) = y and h(x,y) = x, we have

(f ?α g) ?α h = α(α(f ) ? α(g)) ? α(h) 1 1 = α 1 ? y + c ? (ax + b) = α y + c ? (ax + b), a a 116 Chapter 7. Deformation and quantization of Hom-algebras

and

f ?α (g ?α h) = α(α(f )) ? α(α(g) ? α(h)) 1 1 = 1 ? α y + c ? (ax + b) = α y + c ? α(ax + b) a a

which are diﬀerent in general. More generally, we can consider the Poisson algebra of poly- · { } nomials of n > 3 variables (R[x1,...,xn], , , ) where the Poisson { } X ∂f ∂g bracket of two polynomials is given by f ,g = τij , ∂xi ∂xj 16i,j6n × with τ = (τij) an antisymmetric n n real matrix.

The associated associative algebra is (R[x1 ...,xn][[λ]],?) where the ?-product is given by

X X ∂nf ∂ng λn f ? g = σ ···σ , (7.2.7) i1j1 injn ··· ··· ∂xi1 ∂xin ∂xj1 ∂xjn n! n N 16i1,j1,...,in,jn6n ∈ where σ = (σij) is the matrix whose antisymmetrization is τ. For 1 X ∂nf ∂ng n 3, set µ = σ ···σ . > n i1j1 injn ··· ··· n! ∂xi1 ∂xin ∂xj1 ∂xjn 16i1,j1,...,in,jn6n → ◦ ◦ 2 Proposition 7.2.5 A morphism α : R[x1,...,xn] R[x1,...,xn] satisfying α µi = µi α⊗ ∈ for all i N which gives (R[x1,...,xn],?α = α?,α) a structure of Hom- associative algebra is of the form

∀ ∀ − 1 6 i 6 n, α(xi) = xi + bi or 1 6 i 6 n, α(xi) = xi + bi, (7.2.8) ∈ where bi R. Proof. The proof is similar to the case with two variables, we get α(xi) = aixi + bi, except that this time, aiaj = 1 for all i , j, which gives the two cases of the proposition.

To obtain a Hom-associative algebra from (R[x,y][[λ]],?), the ∀ ∈ ◦ ◦ 2 condition i N, α µi = µi α⊗ is very strong. We will see how to construct other morphisms in order to get an Hom-associative algebra by twist. The construction is done by quantization of Pois- son morphisms of R.

7.2.2 Twists of the Poisson bracket We consider again the Poisson algebra of polynomials of two variables (R[x,y],·,{ , }). 7.2. Quantization and twisting of ?-products 117

· ◦ · In order to construct a Hom-Poisson algebra (R[x,y], α = α , { } ◦ { } , α = α , ,α) using the twisting principle Theorem 6.1.4, we need a Poisson morphism α : R[x,y] → R[x,y]. Such a morphism preserve in particular the point-wise multiplication and so is entirely deﬁned by its images on x and y. De- ∈ noting α(x) C φ1(x,y), α(y) C φ2(x,y) where φi R[x,y], applying the Poisson morphism α amounts to precompose by a polynomial 2 → 2 ◦ map φ = (φ1,φ2): R R , i.e. α(f ) = φ∗(f ) = f φ. → Proposition 7.2.6 The map φ∗ : R[x,y] R[x,y] is a Poisson algebra morphism if and only if

∂xφ1 ∂yφ1 − Jφ = = ∂xφ1∂yφ2 ∂xφ2∂yφ1 = 1 (7.2.9) ∂xφ2 ∂yφ2

where Jφ is the jacobian determinant of φ. 2 → 2 ∈ Proof. The map φ∗ is deﬁned by φ = (φ1,φ2): R R . Let f ,g R[x,y]. By deﬁnition, we have

(f · g) ◦ φ = f ◦ φ · g ◦ φ ⇔ φ∗(f · g) = φ∗(f ) · φ∗(g). By linearity, we may assume now that f (x,y) = axkyl and g(x,y) = bxpyq, with a,b ∈ R, k,l,p,q ∈ N. We have { ◦ ◦ } k l p q − k l p q f φ,g φ = ∂x(aφ1φ2)∂y(bφ1φ2) ∂y(aφ1φ2)∂x(bφ1φ2) − k+p 1 l+q 1 − = ab(kq lp)φ1 − φ2 − (∂xφ1∂yφ2 ∂xφ2∂yφ1) and { } ◦ k l p q − k l p q ◦ f ,g φ = ∂x(ax y )∂y(bx y ) ∂y(ax y )∂x(bx y ) φ − k+p 1 l+q 1 = ab(kq lp)φ1 − φ2 − .

By identiﬁcation, φ∗ is a Poisson algebra morphism if and only if Jφ = 1.

Polynomial automorphisms of R2 are known, those of constant jacobian 1 are generated by triangular transformations1.

Theorem 7.2.7 (Automorphisms theorem) Let K be a ﬁeld. The group of polynomial au- 2 tomorphisms Aut[K ] is generated by aﬃne transformations (x,y) 7→ ∈ − (a1x + b1y + c1,a2x + b2y + c2) where ai,bi,ci K with a1b2 a2b1 , 0, and by triangular transformations (x,y) 7→ (x,y+p(x)) where p ∈ K[x]. However, if we don’t assume that the map φ : R2 → R2 is bi- jective, equation Jφ = 1 don’t allows yet to conclude that φ is invertible. On K = C, this is the jacobian conjecture, which still is an 1since their jacobian matrix is triangular 118 Chapter 7. Deformation and quantization of Hom-algebras

open question even for n = 2.

Jacobian conjecture A polynomial map of Cn whose jacobian is a is a non-zero constant is an automorphism of Cn.

The automorphisms theorem was discovered by Jung in 1942 for ﬁelds of characteristic 0 and extended by Van der Kulk for any ﬁeld. Various proofs were proposed, a simple proof on C and references are given in article [VC03], which also mention references on the work on the jacobian conjecture.

Corollary 7.2.8 The Poisson automorphisms group of R = (R[x,y],·,{ , }) is generated by lower triangular transformations (x,y) 7→ (x,y + p(x)) and upper triangular transformations (x,y) 7→ (x + q(y),y), where p ∈ K[x], q ∈ K[y]. Proof. Indeed, we set in matrix form

a b x ax + by : 7→ c d y cx + dy

the linear transformations, and

1 q(y)/y x x + q(y) 1 0 x x : 7→ et : 7→ 0 1 y y p(x)/x 1 y y + p(x)

the upper and lower triangular transformations. Poisson automorphisms being of jacobian 1, for an automorphism which is a product of linear and triangular transformations, the linear transformations must be of jacobian 1 too, since it is already the case for the triangular ones. Hence

a b a b ∈ SL(2, ) = M = ∈ M ( ),detM = ad − bc = 1 . c d R c d 2 R

Since SL(2,R) is generated by transvections, which are exactly the matrix of the triangular morphisms with a polynomial p or q which is linear, the result follows.

7.2.3 Quantization of the Poisson automorphisms

In this section, we set forth the way to quantize Poisson automorphisms of R “à la Fedosov”, (see [Fed96]), to obtain morphisms of the associative Moyal-Weyl algebra. 7.2. Quantization and twisting of ?-products 119

Morphisms and ﬂows

Let φ(x,y) = (x,y + p(x)) be a triangular morphism, with p ∈ R[x]. Up to add a parameter t, we can assume that φt(x,y) = (x,y +tp(x)) is the ﬂow of the hamiltonian equation

 ∂H x˙ = (x,y) = 0 (  ∂y x(t) = x0 ⇔ (7.2.10)  ∂H y(t) = y0 + tp(x) y˙ = − (x,y) = p(x) ∂x −R − with H(x,y) = p(x)dx a primitive of p. So φ0(x0,y0) = (x0,y0) 2 is the identity of R and φt(x0,y0) = (x(t),y(t)) = (x0,y0 + tp(x0)) is the solution at time t. Differentiating with respect to t, we have d φ (x,y) = (0,p(x)) = X (x(t),y(t)) = X ◦ φ (x,y) dt t H H t ∂H ∂H where X = − is the Lie derivative of H. H ∂y ∂x → 2 As a function of time φt : R R , the flow therefore satisfies the differential equation  d  φ = X ◦ φ dt t H t . (7.2.11)  φ0 = idR2 We obtain the same equation for the flow of a triangular morphism (x,y) 7→ (x + tq(y),y) with q ∈ R[y].

Such a differential formal equation is a way to concisely write the hidden recursion satisfied by the coefficients of the solution power series. A differential formal equation of this type admits an unique solution (see [Che61, Theorem 1.1]) given by iterated integrations

! X Z t φt = Qk(t) φ0 with Qk+1(t) = XH (s)Qk(s)ds and Q0 = id (7.2.12) 0 k N ∈ Moreover, if XH (t) = XH is constant in time as it is the case here, we get

tXH φt = e φ0. (7.2.13) Since we work in the polynomial area with successive compositions of diﬀerentials operators, the power series are ﬁnite sums when evaluated on polynomials. 120 Chapter 7. Deformation and quantization of Hom-algebras

◦ Let φt∗(f ) = f φt be the Poisson algebra morphism associated to the ﬂow, we have

∂H ◦ ! d d ∂f ∂f ∂y φt φ∗(f ) = (f ◦φ ) = ◦ φ ◦ φ = {f ,H}◦φ , dt t dt t ∂x t ∂y t − ∂H ◦ t ∂x φt so the Poisson morphism φt∗ is well-deﬁned by  d  − ◦ φt∗ = φt∗ PH { } dt where PH (f ) = H,f . (7.2.14)  φ0∗ = idR 1 ◦ ··· ◦ n More generally, for a Poisson automorphism ψt∗ = φt ∗ φt ∗ i where the φt∗ are triangular morphisms, we obtain

n d X 1 k 1 d k k+1 n ψ∗ = φ ∗ ◦ ··· ◦ φ − ∗ ◦ φ ∗ ◦ φ ∗ ··· ◦ φ ∗ dt t t t dt t t t k=1 n X − 1∗ ◦ ··· ◦ k 1∗ ◦ ◦ k+1∗ ··· ◦ n = φt φt − PHk φt φt ∗ k=1 n X 1 − ◦ k+1∗ ··· ◦ n − ◦ ◦ k+1∗ ··· ◦ n = ψt∗ φt φt ∗ PHk φt φt ∗ | {z } k=1 >k φt ∗ where P (f ) = {H ,f } with the various hamiltonians H corre- Hk k k sponding to the triangular morphisms φk. Since

1 1 n o >k∗ − ◦ ◦ >k∗ >k∗ − >k∗ φt PHk φt (f ) = φt Hk,φt (f ) 1 >k − = φt ∗ (Hk),f

P k (f ), C Ht

n X setting P = P k , the Poisson morphism ψ is well-deﬁned by Ht Ht t∗ k=1  d  ψ∗ = −ψ∗ ◦ P dt t t Ht , (7.2.15)  ψ0∗ = idR here the operator PHt depend on the time t. We can recover the fact that ψt∗ is a morphism from R by diﬀer- entiation. Let

⊗ { } − { } At(f g) B ψt∗( f ,g ) ψt∗(f ),ψt∗(g) . 7.2. Quantization and twisting of ?-products 121

Then

d d A = ψ∗({f ,g}) − {ψ∗(f ),ψ∗(g)} dt t dt t t t − { { }} { { } } = ψt∗( Ht, f ,g ) + ψt∗( Ht,f ),ψt∗(g) { { } } + ψt∗(f ),ψt∗( Ht,g ) − { } }} { { } } = ψt∗( Ht,f ,g ) + ψt∗( Ht,f ),ψt∗(g) − { { }} { { } } ψt∗( f , Ht,g ) + ψt∗(f ),ψt∗( Ht,g ) − { } ⊗ − ⊗ { } = At( Ht,f g) At(f Ht,g ) so At satisﬁes the diﬀerential equation d A = −A ◦ P ⊗ id + id ⊗ P dt t t Ht Ht with the initial condition A0 = 0. Thus the unique solution is At = A0 = 0, so we have that ψt∗ preserve the Poisson bracket.

Quantiﬁcation In order to quantize Poisson automorphisms to obtain morphisms 2 { } of the associative algebra (R[x,y][[λ]],?), we replace PHt = Ht, 1 in equation (7.2.15) by Q : f 7→ (H ? f − f ? H ). Ht λ t t As in (7.2.12), solutions of

 d  α = −α ◦ Q dt t t Ht (7.2.16)  α0 = id write as formal power series

Z t Z t1 Z tn 1 X∞ − n ··· − ◦ ◦ ··· ◦ αt = id+ ( 1) QH dtn QH dtn 1 QH dt1. tn tn 1 − t1 n=1 0 0 0 −

To show that they are morphisms of (R[x,y][[λ]],?) i.e. they preserve the product ?, set ⊗ − Bt(f g) B αt(f ? g) αt(f ) ? αt(g).

Then d d B = α (f ? g) − α (f ) ? α (g) dt t dt t t t − ◦ ◦ ◦ = αt QHt (f ? g) + αt QHt (f ) ? αt(g) + αt(f ) ? αt QHt (g) − ⊗ − ⊗ = Bt(QHt (f ) g) Bt(f QHt (g))

2here ? denote the Moyal-Weyl ?-product or its equivalent simpler form 122 Chapter 7. Deformation and quantization of Hom-algebras so Bt satisﬁes the diﬀerential equation d B = −B ◦ Q ⊗ id + id ⊗ Q dt t t Ht Ht with the initial condition B0 = 0. Thus the unique solution is Bt = B0 = 0, showing that αt is a morphism of (R[x,y][[λ]],?).

In particular, if we take only one triangular morphism φ (x,y) = R t (x,y + tp(x)) with p ∈ R[x] of degree d and H(x,y) = − p(x)dx the corresponding hamiltonian, QHt = QH doesn’t depend on t. Since H is a function of x only, for the Moyal-Weyl ?-product (7.2.2), we have

l d+1 m 2 1 X 1 λ2k Q (f ) = (H? f −f ? H) = − p(2k)(x)∂2k+1(f ), H λ MW MW (2k + 1)! 2 y k=0 and for the ?-product (7.2.3),

d+2 n 1 X λ (n 1) n Q (f ) = (H ? f − f ? H) = − p − (x)∂ (f ). H λ n! y n=1

The operator QH doesn’t depend on t, so the diﬀerential equation (7.2.16) integrates in tQH αt = e− , which is a ﬁnite sum when evaluated on a polynomial f , because n the compositions QH only involve successive partial derivations with respect to y of f . Therefore, we obtain other examples of morphisms αt of (R[x,y][[λ]],?) than those of Proposition 7.2.4, and they can be used to obtain a

Hom-associative algebra (R[x,y][[λ]],?αt ,α) by the twisting principle.

A Computations with Mathemat- ica

Contents A.1 Computation of the morphisms of sl(2) ...... 126 A.2 Hom-Lie structures associated to the Jackson sl(2) bracket ...... 129

e give here Mathematica commands which were used to carry W out various computations to obtain Hom-Lie algebras.

125 126 Appendix A. Computations with Mathematica

A.1 Computation of the morphisms of sl(2)

Let sl(2) be the Lie algebra generated by {e,f ,h} with the bracket given by [h,e] = 2e,[h,f ] = −2f ,[e,f ] = h. We search the morphisms α : sl(2) → sl(2) satisfying α([x,y]) = { } [α(x),α(y)]. We write α = aij in the basis e,f ,h . This method was used in [Yau11].

We deﬁne the Lie bracket so it satisﬁes bilinearity properties. Croch[f_ + g_,h_]:=Croch[f ,h] + Croch[g,h] Croch[f_,g_ + h_]:=Croch[f ,g] + Croch[f ,h] Croch[t_Real f_,g_]:=tCroch[f ,g] Croch[t_Rational f_,g_]:=tCroch[f ,g] Croch[t_Integer f_,g_]:=tCroch[f ,g] Croch[f_,t_Real g_]:=tCroch[f ,g] Croch[f_,t_Rational g_]:=tCroch[f ,g] Croch[f_,t_Integer g_]:=tCroch[f ,g] {Croch[a[i_,j_]f_,g_]:=a[i,j]Croch[f ,g],Croch[f_,a[i_,j_]g_]:=a[i,j]Croch[f ,g]} {Null,Null}

We give the values of the bracket on the basis. {Croch[e,e] = 0,Croch[e,f ] = h,Croch[e,h] = −2e, Croch[f ,e] = −h,Croch[f ,f ] = 0,Croch[f ,h] = 2f , Croch[h,e] = 2e,Croch[h,f ] = −2f ,Croch[h,h] = 0} {0,h,−2e,−h,0,2f ,2e,−2f ,0}

We deﬁne linearity and multiplicativity properties for the morphism α. alpha[0]:=0;alpha[1]:=1; alpha[x_ + y_]:=alpha[x] + alpha[y] alpha[x_y_]:=alpha[x]alpha[y] alpha[x_∧n_Integer]:=alpha[x]∧n alpha[t_Real x_]:=talpha[x] alpha[t_Rational x_]:=talpha[x] alpha[t_Integer x_]:=talpha[x]

We deﬁne the morphism α by a matrix. A = Array[a,{3,3}] {{a[1,1],a[1,2],a[1,3]},{a[2,1],a[2,2],a[2,3]},{a[3,1],a[3,2],a[3,3]}} MatrixForm[A]  a[1,1] a[1,2] a[1,3]   a[2,1] a[2,2] a[2,3]  a[3,1] a[3,2] a[3,3] We compute the images of α on the basis. X = {e,f ,h} A.1. Computation of the morphisms of sl(2) 127

{e,f ,h} For[i = 1,i < 4,i++,alpha[X[[i]]] = (Transpose[A].X)[[i]]] i=. {alpha[e],alpha[f ],alpha[h]} {ea[1,1]+f a[2,1]+ha[3,1],ea[1,2]+f a[2,2]+ha[3,2],ea[1,3]+f a[2,3]+ ha[3,3]} MatrixForm[%]  ea[1,1] + f a[2,1] + ha[3,1]   ea[1,2] + f a[2,2] + ha[3,2]  ea[1,3] + f a[2,3] + ha[3,3] We write the polynomials in e,f ,h obtained by computation of α([x,y]) − [α(x),α(y)] on the basis. We only need three of this polynomials: α([e,f ])−[α(e),α(f )] , α([e,h])−[α(e),α(h)] , α([f ,h])− [α(f ),α(h)]. poly = SparseArray[{{i_,j_} → If[i>=j,0, Collect[Expand[Croch[alpha[X[[i]]],alpha[X[[j]]]] − alpha[Croch[X[[i]],X[[j]]]]], {e,f ,h}]]},{3,3}] SparseArray[< 3 >,{3,3}] Normal[poly] {{0,e(−a[1,3]+2a[1,2]a[3,1]−2a[1,1]a[3,2])+f (−a[2,3]−2a[2,2]a[3,1]+ 2a[2,1]a[3,2])+h(−a[1,2]a[2,1]+a[1,1]a[2,2]−a[3,3]),h(−a[1,3]a[2,1]+ a[1,1]a[2,3] + 2a[3,1]) + e(2a[1,1] + 2a[1,3]a[3,1] − 2a[1,1]a[3,3]) + f (2a[2,1]−2a[2,3]a[3,1]+2a[2,1]a[3,3])},{0,0,h(−a[1,3]a[2,2]+a[1,2]a[2,3]− 2a[3,2]) + e(−2a[1,2] + 2a[1,3]a[3,2] − 2a[1,2]a[3,3]) + f (−2a[2,2] − 2a[2,3]a[3,2] + 2a[2,2]a[3,3])},{0,0,0}}

To cancel these polynomials, we extract their coefficients in e,f ,h and solve the equations we get, the unknowns being the coefficients aij of the morphism α. For[i = 1,i < 4,i++, For[j = 1,j < 4,j++, If[i < j,coefs[i,j] = CoefficientList[poly[[i,j]],{e,f ,h}],0] ] ] vars = Normal[Flatten[A]] {a[1,1],a[1,2],a[1,3],a[2,1],a[2,2],a[2,3],a[3,1],a[3,2],a[3,3]} eqs = DeleteCases[Flatten[{coefs[1,2],coefs[1,3],coefs[2,3]}],0] {−a[1,2]a[2,1]+a[1,1]a[2,2]−a[3,3],−a[2,3]−2a[2,2]a[3,1]+2a[2,1]a[3,2],−a[1,3]+ 2a[1,2]a[3,1]−2a[1,1]a[3,2],−a[1,3]a[2,1]+a[1,1]a[2,3]+2a[3,1],2a[2,1]− 2a[2,3]a[3,1]+2a[2,1]a[3,3],2a[1,1]+2a[1,3]a[3,1]−2a[1,1]a[3,3],−a[1,3]a[2,2]+ a[1,2]a[2,3]−2a[3,2],−2a[2,2]−2a[2,3]a[3,2]+2a[2,2]a[3,3],−2a[1,2]+ 2a[1,3]a[3,2] − 2a[1,2]a[3,3]} zeros = Table[0,{i,Length[eqs]}] {0,0,0,0,0,0,0,0,0} sols = Solve[eqs == zeros,vars] 128 Appendix A. Computations with Mathematica

Solve::svars : Equations may not give solutions for all "solve" variables.ii nn 2 → − a[3,1] → 2a[3,1] → 1 → a[1,1] a[2,1] ,a[1,3] a[2,1] ,a[1,2] a[2,1] ,a[2,3] 0 , a[3,3] → −1,a[2,2] → 0,a[3,2] → 0}, n 2 → 1 → → → − a[2,3] a[1,1] a[2,2] ,a[1,3] 0,a[1,2] 0,a[2,1] 4a[2,2] , o → − a[2,3] → → a[3,1] 2a[2,2] ,a[3,3] 1,a[3,2] 0 , n 2 2 → 1+2a[3,3]+a[3,3] → a[3,2] a[3,2]a[3,3] → − a[3,2] a[1,1] 4a[2,2] ,a[1,3] − a−[2,2] ,a[1,2] a[2,2] , 2 a[2,2](1 2a[3,3]+a[3,3] ) 1 a[3,3]2 a[2,2]( 1+a[3,3]) → − − → → a[2,1] 4a[3,2]2 ,a[3,1] −4a[3,2] ,a[2,3] a−[3,2] , {a[1,1] → 0,a[1,3] → 0,a[1,2] → 0,a[2,1] → 0,a[3,1] → 0, a[2,3] → 0,a[3,3] → 0,a[2,2] → 0,a[3,2] → 0}}

The ﬁtting morphisms α are of the following form. MatrixForm[A]/.sols   1   [3 1]2 2 [3 1]  0 0  − a , 1 a , a[2,2]  a[2,1] a[2,1] a[2,1]  a[2,3]2   [2 1] 0 0 , − a[2,2] a[2,3]  ,  a ,   4a[2,2]   a[3,1] 0 −1 − a[2,3]  2a[2,2] 0 1  1+2a[3,3]+a[3,3]2 a[3,2]2 a[3,2] a[3,2]a[3,3]   − − −   4a[2,2] a[2,2] a[2,2] 0 0 0   a[2,2](1 2a[3,3]+a[3,3]2) a[2,2]( 1+a[3,3])    − − , 0 0 0  4a[3,2]2 a[2,2] a−[3,2]     2   1 a[3,3] 0 0 0  −4a[3,2] a[3,2] a[3,3]  These morphisms indeed are invertible since they have non- vanishing determinant, moreover, this determinant is 1 (except for the zero morphism). Simplify[Det[A]/.sols] {1,1,1,0}

We rename the parameters. Part[MatrixForm[A]/.sols,1]/.{{1/a[2,1] → λ,a[2,1] → 1/λ,a[3,1] → µ}}  −λµ2 λ 2λµ   1   λ 0 0   µ 0 −1  Part[MatrixForm[A]/.sols,2]/.{{1/a[2,2] → λ,a[2,2] → 1/λ,a[2,3] → µ}}    λ 0 0   2   − λµ 1   4 λ µ   − λµ  2 0 1 Part[MatrixForm[A]/.sols,3]/.{{1/a[2,2] → λ,a[2,2] → 1/λ,a[3,3] → µ,a[3,2] → ν}}  1 2 − 2 − −   4 λ 1 + 2µ + µ λν λ( ν µν)   1 2µ+µ2 1 1+µ   − − −   4λν2 λ λν   1 µ2  4−ν ν µ A.2. Hom-Lie structures associated to the Jackson sl(2) bracket 129

These matrices correspond to automorphisms sl(2) → sl(2), 1 X 7→ A− .X.A with 0 b A = with λ = −c2, µ = 2bd for the ﬁrst one, −1/b d a b A = with λ = d2, µ = −ab for the second one, 0 1/a a b 2 − A = 1+bc with λ = 1/a , µ = 1 + 2bc, ν = 2ac for the third one. c a A.2 Hom-Lie structures associated to the Jack- son sl(2) bracket

We obtain the Lie Jackson sl(2) algebra by deformation. It is generated by {e,f ,h} with bracket given by [h,e] = 2e,[h,f ] = −2(1 + t t)f ,[e,f ] = 1 + 2 h. We deﬁne this new bracket as before so that it satisﬁes bilinearity properties. Croch[f_ + g_,h_]:=Croch[f ,h] + Croch[g,h] Croch[f_,g_ + h_]:=Croch[f ,g] + Croch[f ,h] Croch[t_Real f_,g_]:=tCroch[f ,g] Croch[t_Rational f_,g_]:=tCroch[f ,g] Croch[t_Integer f_,g_]:=tCroch[f ,g] Croch[f_,t_Real g_]:=tCroch[f ,g] Croch[f_,t_Rational g_]:=tCroch[f ,g] Croch[f_,t_Integer g_]:=tCroch[f ,g] {Croch[a[i_,j_]f_,g_]:=a[i,j]Croch[f ,g],Croch[f_,a[i_,j_]g_]:=a[i,j]Croch[f ,g]} {Null,Null} Croch[tf_,g_]:=tCroch[f ,g] Croch[f_,tg_]:=tCroch[f ,g]

We give the values of the bracket on the basis. {Croch[e,e] = 0,Croch[e,f ] = (1 + t/2)h,Croch[e,h] = −2e, Croch[f ,e] = −(1 + t/2)h,Croch[f ,f ] = 0,Croch[f ,h] = 2(1 + t)f , Croch[h,e] = 2e,Croch[h,f ] = −2(1 + t)f ,Croch[h,h] = 0} t − − − t − 0,h 1 + 2 , 2e,h 1 2 ,0,2f (1 + t),2e, 2f (1 + t),0

We search linear maps α which satisfies Hom-Jacobi identity for this bracket. We define the linearity properties for α. alpha[0]:=0;alpha[1]:=1; alpha[x_ + y_]:=alpha[x] + alpha[y] alpha[t_Real x_]:=talpha[x] alpha[t_Rational x_]:=talpha[x] alpha[t_Integer x_]:=talpha[x] 130 Appendix A. Computations with Mathematica alpha[tf_]:=talpha[f ] We define the map α by a matrix. A = Array[a,{3,3}] {{a[1,1],a[1,2],a[1,3]},{a[2,1],a[2,2],a[2,3]},{a[3,1],a[3,2],a[3,3]}} MatrixForm[A]  a[1,1] a[1,2] a[1,3]   a[2,1] a[2,2] a[2,3]  a[3,1] a[3,2] a[3,3] X = {e,f ,h} {e,f ,h} For[i = 1,i < 4,i++,alpha[X[[i]]] = (Transpose[A].X)[[i]]] i=. {alpha[e],alpha[f ],alpha[h]} {ea[1,1]+f a[2,1]+ha[3,1],ea[1,2]+f a[2,2]+ha[3,2],ea[1,3]+f a[2,3]+ ha[3,3]} MatrixForm[%]  ea[1,1] + f a[2,1] + ha[3,1]   ea[1,2] + f a[2,2] + ha[3,2]  ea[1,3] + f a[2,3] + ha[3,3] We have to find the coefficients of α such that the Hom-Jacobi identity is satisfied. Jac = Croch[alpha[e],Croch[f ,h]] +Croch[alpha[f ],Croch[h,e]] +Croch[alpha[h],Croch[e,f ]] − − t t 2h 1 2 a[2,2]+4ea[3,2]+a[1,3]Croch e,h 1 + 2 +2a[1,1]Croch[e,f (1+ t t t)]+a[2,3]Croch f ,h 1 + 2 +2a[2,1]Croch[f ,f (1+t)]+a[3,3]Croch h,h 1 + 2 + 2a[3,1]Croch[h,f (1 + t)]

We extract coefficients of this polynomial in e,f ,h and solve the corresponding system to cancel them, unknown being the coefficients aij of the map α. Collect[ExpandAll[Jac],{e,f ,h}] h2a[1,1] + 3ta[1,1] + t2a[1,1] − 2a[2,2] − ta[2,2] +f 2a[2,3] + 3ta[2,3] + t2a[2,3] − 4a[3,1] − 8ta[3,1] − 4t2a[3,1] +e(−2a[1,3] − ta[1,3] + 4a[3,2]) eqs = CoefficientList[Collect[ExpandAll[Jac],{e,f ,h}],{e,f ,h}] 0,2a[1,1] + 3ta[1,1] + t2a[1,1] − 2a[2,2] − ta[2,2] , 2a[2,3] + 3ta[2,3] + t2a[2,3] − 4a[3,1] − 8ta[3,1] − 4t2a[3,1],0 , {{−2a[1,3] − ta[1,3] + 4a[3,2],0},{0,0}}} equas = DeleteCases[Flatten[eqs],0] 2a[1,1] + 3ta[1,1] + t2a[1,1] − 2a[2,2] − ta[2,2] , 2a[2,3] + 3ta[2,3] + t2a[2,3] − 4a[3,1] − 8ta[3,1] − 4t2a[3,1], −2a[1,3] − ta[1,3] + 4a[3,2]} vars = Flatten[A] A.2. Hom-Lie structures associated to the Jackson sl(2) bracket 131

{a[1,1],a[1,2],a[1,3],a[2,1],a[2,2],a[2,3],a[3,1],a[3,2],a[3,3]} sols = Solve[equas == {0,0,0},vars] Solve::svars : Equations may not give solutions for all "solve" variables.ii nn oo → − 1 − − → − − − → 4(1+t)a[3,1] a[3,2] 4 ( 2 t)a[1,3],a[2,2] ( 1 t)a[1,1],a[2,3] 2+t We rename the parameters. MatrixForm[A]/.sols/.{a[1,1] → a,a[2,1] → b,a[3,1] → c, a[1,2] → d,a[1,3] → k,a[3,3] → l,t → q − 1}//Simplify   a d k    4cq   b aq 1+q   1  c 4 k(1 + q) l Specifying coeﬃcients, we recover the twists corresponding to the already known deformations. MatrixForm[A]/.sols/.{a[1,1] → 1 + t,a[2,1] → 0,a[3,1] → 0, a[1,2] → 0,a[1,3] → 0,a[3,3] → 1 + t}//Simplify    1 + t 0 0   0 (1 + t)2 0   0 0 1 + t  MatrixForm[A]/.sols/.{a[1,1] → (2 + t)/(2(1 + t)),a[2,1] → 0,a[3,1] → 0, a[1,2] → 0,a[1,3] → 0,a[3,3] → 1}//Simplify  2+t  2+2t 0 0  t   0 1 + 2 0   0 0 1  132 Appendix A. Computations with Mathematica Bibliography

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