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study of Hom-(co)Poisson algebras

Olivier Elchinger

To cite this version:

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�

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 byMartinBordemannand AbdenacerMakhlouf defended the 12^thof November 2012 in front of the jury composed by:

M. BenjaminEnriquez Université de Strasbourg (Referee)

M. JoakimArnlind University of Linköping (Referee)

M. CamilleLaurent-Gengoux Université de Metz (Referee)

M. MartinSchlichenmaier Université de Luxembourg

M. MartinBordemann Université de Haute Alsace (Advisor)

M. AbdenacerMakhlouf Université de Haute Alsace (Advisor)

M. AugustinFruchard Université de Haute Alsace (Invited member)

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 M^meWurth 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 M^me Fricker, pour la gestion et résolution d’innombrables détails ad- ministratifs, M^me 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

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

. . . 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 algebraso(3) . . . 66

4.2 Subalgebra of the Chevalley-Eilenberg complex . 67 4.3 Deformation retract of the complex . . . 70

4.4 Computation of theL∞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

7 Deformation and quantization of Hom-algebras

₁₀₉

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 ofsl(2) . . . 126 A.2 Hom-Lie structures associated to the Jacksonsl(2)

bracket . . . 129

Bibliography

₁₃₃

vii

T

^his thesis aims to study some algebraic aspects of structures linked to the problem of deformation quantization. At first, we examine the formality for the case of free algebras and for the Lie algebraso(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 quan- tum physics. If one has a quantum description of a physical sys- tem, 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 calledquanti- zation. 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 quan- tum 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

the algebra of smooth functions over a Poisson manifold admits a graded Lie algebraic structure by shift, which controls the defor- mations of the Poisson bracket. Kontsevich shows that this com- plex is linked with its cohomology — which therefore controls the same deformations — by a L∞-quasi-isomorphism, called a for- mality 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 al- gebra. 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 uni- versal 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 for- mality in general, except in the trivial cases. The study of the Lie algebraso(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 alge- bras and in degrees 0 and 3 for the Lie algebra so(3). We then build aL∞-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 per- turbation 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 theq-deformations of Witt and Virasoro algebra, leads to the introduction of new non-associative structures. Hom-Lie al- gebras were first introduced by Hartwig, Larsson and Silvestrov in order to describe theses q-deformations using σ-derivations (see [HLS06]). They describe theq-deformation of the Witt algebra by a one-parameter q family of Hom-Lie algebras, such that the ini- tial Witt algebra is obtained forq= 1. The associative type objects x

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 al- gebraic structures, and links between cohomology and formal de- formation 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 princi- ple by twist to build new structures of the same type, or to de- form 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 envelop- ing algebras from Hom-Lie algebras endowed with a Hom-coPois- son structure and Hom-Lie bialgebras.

xi

Formality and L ^∞ structures

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 algebraso(3) . . . 66

4.2 Subalgebra of the Chevalley-Eilenberg complex . 67 4.3 Deformation retract of the complex . . . 70

4.4 Computation of theL∞structure . . . 71

Introduction of the first part

I

ⁿthis first part, after presenting graded structures in Section 1.1 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). AL∞-algebra is a generalization of a graded Lie al- gebra. 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 com- plex and its cohomology which, endowed with the Gerstenhaber bracket and the induced Schouten bracket, become graded Lie al- gebras by shift. They can be considered asL∞-algebras, differen- tials 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 isomor- phism 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 perturbedL∞ structure gives only one more map of arity 3.

The same work is done for the Lie algebraso(3). Again, there is no formality (Theorem 4.4.1) and the modifiedL∞ structure consists of only one term of arity 3 too.

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

T

^o begin, we expose the workplace, which is the category of 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

1.1 Graded structures

A gradation on a space enable the concise exposition of valid prop- erties in each degree. Computations are also easier by working with homogeneous elements in each component. In the following, Kis 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- vector spaces Z-graded V =⊕_i∈

ZVⁱ, direct sum of subspaces Vⁱ. An element x of V lying in one of the Vⁱ 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 spacesV andW, a linear mapφ:V →W is said to be homogeneous of degree j if and only if for all inte- gers i we have φ(Vⁱ)⊂W^i+j. In graded situations we shall write Hom(V , W)^j for the vector space of all linear maps which are ho- mogeneous 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 ⊗ W)ⁱ =⊕_k∈

ZV^k⊗W^i−k, see [ML63]. The tensor product of two mor- phismφ:V →W andψ:V⁰→W⁰ is defined with theKoszul rule of signs: for all homogeneous elementsx∈V andy∈V⁰

(φ⊗ψ)(x⊗y)B(−1)^|ψ||x|φ(x)⊗ψ(y), (1.1.1) withψof degree|ψ|. There also is thegraded transposition

τ:V ⊗W →W ⊗V

τ(x⊗y)B(−1)^|x||y|y⊗x. (1.1.2) Theses two rules will determine all the signs which will appears

in computations.

A graded (associative) algebra (A, µ) is a graded vector spaceA together with graded space morphism µ:A ⊗ A → Aof degree 0 satisfying

AⁱA^j ⊂ A^i+j.

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

(a⊗b)(a⁰⊗b⁰)B(−1)^|b||a0|aa⁰⊗bb⁰.

A graded algebra (A, µ) iscommutativeifµ◦τ=µ,anticommutative ifµ◦τ=−µ; fora, b∈ Athis writes

ab= (−1)^|a||b|ba et ab=−(−1)^|a||b|ba.

A graded coalgebra(C,∆) is defined in a similar way: the comulti- plication must satisfy

∆C^j ⊂ X

k+l=j

C^k⊗ C^l.

It is saidcocommutativeifτ◦∆=∆andanticocommutativeifτ◦∆=

−∆. 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 morphismd:A → Aof degreei such that

d(ab) =da.b+ (−1)^i|a|a.db which also writes

d◦µ=µ◦(d⊗idA+idA⊗d).

Acoderivation of degreei in a graded coalgebra (C,∆) is a linear morphismd:C → C of degreei such that if we note∆a=P

(a)a₁⊗ a₂, we have

∆da=X

(a)

da1⊗a2+ (−1)^i|a1|a1⊗da2

or also

∆◦d = (d⊗idC+idC⊗d)◦∆.

Definition 1.1.1 A graded Lie algebrais aZ-graded vector spaceV endowed with a graded Lie bracket,i.e. a bilinear map [, ] :V ⊗V →V such that gradation [Vⁱ, V^j]⊂V^i+j

and forx, y, z∈V graded antisymmetry

[y, x] =−(−1)^|y||x|[x, y] (1.1.3) graded Jacobi identity

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

x,y,z(−1)^|x||z|[[x, y], z] = 0,

where _x,y,z indicates a summation on cyclic permutations ofx, y, z.

1.1.2 Shifted spaces

For an integer j denote by V[j] the shifted graded vector spacede- fined byV[j]ⁱ BV^i+j. The identity mapV →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 n^th power of the suspension mapsBs¹:V[j]→V[j−1] of degree one.

In particular,s:V[1]→V CV[0] and if an elementx∈V[1] is of degree|x|(inV[1]), thenx=sx is of degree|sx|=|x|+ 1 inV.

The suspension will be ‘visible’ forshifted multilinear maps: let φ : V^⊗k → W^⊗l be a multilinear map of degree |φ|. The shifted map φ[j] : V[j]^⊗k → W[j]^⊗l is defined by setting φ[j] B(s^⊗l)^−j ◦ φ◦(s^⊗k)^j. The degree of the shifted mapφ[j] is given by |φ[j]|= j(k−l) +|φ| and we have (φ[j])[j⁰] = φ[j+j⁰]. Note that (s^⊗k)^j = (−1)^k(k

−1 2

j(j−1)

2 (s^j)^⊗k. In order to compute a shifted map, first write for ξ B x₁⊗ · · · ⊗x_k ∈ V^⊗k the value of φ(ξ) with the Sweedler notation asP

φ₍₁₎(ξ)⊗ · · · ⊗φ_(l)(ξ), withφ_(i)(ξ)∈W. By the Koszul rule of signs (1.1.1), the value of the shifted mapφ[j] onηBy1⊗

· · · ⊗y_k ∈V[j]^⊗k is computed as follows, with ˜ηBs^j(y₁)⊗ · · · ⊗s^j(y_k):

φ[j](y₁⊗ · · · ⊗y_k) = (−1)^k(k

−1) 2

j(j−1)

2 +^l(l−₂^1)−j(−₂^j−1)(−1)^{j (k−1)|y1|+(k−2)|y2|+···+(k−(k−1))|yk|} X(−1)^{j (l−1)|φ(1)( ˜η)|+(l−2)|φ(2)( ˜η)|+···+(l−(l−1))|φ(l)( ˜η)|}

φ₍₁₎(y₁⊗ · · · ⊗y_k)⊗ · · · ⊗φ_(l)(y₁⊗ · · · ⊗y_k).

1.1.3 Tensor bialgebra

For a Z-graded vector space V, we denote by TV =⊕_k∈

NV^⊗k the tensorial algebra overV. It is a K-associative graded algebra with unit1, it inherits itsZ-grading by theZ-grading ofV, see [ML63].

To avoid any confusion, the symbol ⊗ is not written for the free multiplicationµ=µTV inTV, which is the juxtaposition.

Moreover, TV is a graded bialgebra: let TV⁺ = ⊕_k∈

N^?V^⊗k be the augmentation ideal. The counit ε = εTV : TV → K is de- fined by the condition Kerε = TV⁺ and ε(1) = 1_K. The graded comultiplication shuffle ∆_sh is the morphism of associative algebra TV → TV ⊗ TV induced (by universal property Theorem 1.1.2) by its value∆_sh(x) =x⊗1+1⊗xon generatorsx∈V.

Since the multiplicationµ^[2]onTV⊗TV is given by (µ⊗µ)◦(id⊗ τ⊗id) with the graded transposition, there are signs in formulas involving∆_sh, for example∆_sh(xy) =xy⊗1+x⊗y+ (−1)^|x||y|y⊗x+ 1⊗xy, for x ∈ V^|x| and y ∈ V^|y|. This comultiplication is graded cocommutative (i.e. τ◦∆_sh=∆_sh) of degree 0.

Dualizing this bialgebra structure, we obtain on the spaceTV another structure of bialgebra, with comultiplication (not graded cocommutative) of deconcatenation, ∆ = ∆TV which dualize the free multiplication and is given by formula∆(x₁· · ·x_k) =1⊗x₁· · ·x_k+ P_k

r=2x₁· · ·x_r−1⊗x_r· · ·x_k+x₁· · ·x_k⊗1; andmultiplication shuffleµ_sh graded commutative, sometimes written •, which dualize the co- multiplication shuffle. For an explicit formula ofµ_sh, see equation (1.1.9). The two operations ∆andµ_sh are of degree 0, the unit is again1and the counit ε. We notepr_V the canonical projection on TV¹.

For two linear maps ψ₁, ψ₂ from a graded coassociative coal- gebra (C,∆_C) to a graded associative algebra (A, µ_A), the convolu- tion ψ₁? ψ₂ of ψ₁ and ψ₂ with respect to µ_A and ∆_C is given by ψ₁? ψ₂Bµ_A◦(ψ₁⊗ψ₂)◦∆_C. It is a graded associative multiplica- tion in Hom(C, A).

The tensorial algebra (TV , µ=µTV,1) isfree¹ in the sense that it is characterized up to isomorphism by the following universal property.

Theorem 1.1.2 Let(A, µ_A)be an associative graded algebra. Each morphsimφ:V →A of graded vector spaces, of degree zero, extends to a unique morphism φ:T V →Aof graded algebras.

TV ^{φ //}(A, µ_A)

P0V

aa

φ

;; φ satisfies for eachn∈N φ(x₁· · ·x_n) =φ(x₁)· · ·φ(x_n)

⇔φ◦µ⁽ⁿ⁻¹⁾=µ⁽ⁿ_A⁻¹⁾◦(φ⊗ · · · ⊗φ)

(1.1.5)

Proof. The proof given here presents a construction slightly more explicit than the usual one using the universal property of the ten- sorial product.

Let φ^?n = µ⁽ⁿ_A⁻¹⁾◦φ^⊗n◦∆⁽ⁿ⁻¹⁾, with φ^?0 = 1_AεTV and φ^?1 = φ, and let φ = P

n∈Nφ^?n. Extending φ by 0 on ⊕_n,1V^⊗n, we have φ^⊗k+1∆^(k)(x₁· · ·x_n) = 0 for k > n so φ is well-defined. Thus, in the sum defining φ(x₁· · ·x_n), the only remaining term is exactly φ(x1)· · ·φ(xn). Soφis uniquely determined byφ.

The algebra morphismφinduced byφ:V →Ais computed as φ=P

n∈Nφ^?n, the geometric serie using the convolution with re- spect to the multiplicationµ_Aand the comultiplication of decon- catenation∆.

Proposition 1.1.3 Letd :V →Abe a linear map of degreej ∈Z. There exists a unique graded derivation of degreej alongφ, notedd :TV →A, i.e. d◦µ_A=

1and so often called free algebra

µ_A◦(d⊗φ+φ⊗d), such thatd|_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 ofTV

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 allx, y, z∈V, the proposition follows.

This derivation induced byd is computed asφ ? d ? φ. In term of elements, forx₁· · ·x_k ∈ TV, it writes

d(x₁· · ·x_k) =

k

X

r=1

φ(x₁)· · ·φ(x_r−1)d(x_r)φ(x_r+1)· · ·φ(x_k).

A graded coalgebra (C,∆_C, ε_C,1_C) is said to beaugmentedifCis the direct sumC=K1C⊕KerεC. The subspaceC⁺B Ker εCis iso- morphic to the graded quotient coalgebra C/K1_C whitout counit (K1_C 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 integerN such that theN^thiteration of the comultipli- cation vanishes onx. Augmented graded coalgebras (C,∆_C, ε_C,1_C) whose C⁺ (seen as a quotient) are nilpotent form a subcategory C_AN of the category of graded coalgebras. The category C_AN is closed under tensorial product and contains T V. The coalgebra (TV ,∆=∆TV, ε) is cofree in the category C_AN in the sense that it satisfies the following universal property.

Theorem 1.1.4 For each coalgebra (C,∆_C, ε_C)∈ C_AN and each linear map φ:C →V of degree0vanishing on1_C, there exists a unique morphism of graded coalgebrasφ:C→ TV such thatpr_V◦φ=φ.

TV

pr_V !!!!

(C,∆_C)

oo φ

{{ φ

V

φ satisfies ∆◦φ=φ⊗φ◦∆_C

⇔ X

(φ(x))

φ(x)₁⊗φ(x)₂=X

(x)

φ(x₁)⊗φ(x₂) (1.1.6)

Proof. Letφ^?n =µ⁽ⁿ⁻¹⁾◦φ^⊗n◦∆⁽ⁿ_C⁻¹⁾, withφ^?0 =1εC andφ^?1=φ, and let φ = P

n∈Nφ^?n, it is well-defined by nilpotency. For x ∈ K1_C, all equations vanish. For x ∈ Kerε_C, we have, in Sweedler

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) +1ε_C(φ(x₁))⊗φ(x₂)φ(x₃)

+φ(x₁)⊗φ(x₂)

+φ(x₁)φ(x₂)⊗1ε_C(φ(x₃)) +· · ·

which are equal because of the properties of the counit ε_C. For a more rigorous calculus, one can show by induction that forn∈N,

∆◦µ⁽ⁿ⁾=

n

X

i=0

(µ⁽ⁱ⁾⊗µ⁽ⁿ⁻ⁱ⁾)◦(id^⊗i⊗∆⊗id^⊗n−i)−

n−1

X

i=0

µ⁽ⁱ⁾⊗µ^(n−1−i)

(whitµ⁽⁰⁾=id) and then compute

∆◦φ=∆◦X

n∈N

φ^?n=X

n∈N

∆◦µ⁽ⁿ⁻¹⁾◦φ^⊗n◦∆⁽ⁿ_C⁻¹⁾

= X

n∈N n−1

X

i=0

(µ⁽ⁱ⁾⊗µ^(n−1−i))◦(id^⊗i⊗∆⊗id^⊗n−1−i)

−

n−2

X

i=0

µ⁽ⁱ⁾⊗µ^(n−2−i)

!

◦φ^⊗n◦∆⁽ⁿ_C⁻¹⁾ since Imφ⊂V,∆◦φ=φ⊗1+1⊗φ

= X

n∈N n−1

X

i=0

(µ⁽ⁱ⁾⊗µ^(n−1−i))◦(φ^⊗i⊗(φ⊗1+1⊗φ)⊗φ^⊗n−1−i

−

n−2

X

i=0

µ⁽ⁱ⁾⊗µ^(n−2−i)◦φ^⊗n

!

◦∆⁽ⁿ_C⁻¹⁾

= X

n∈N n−1

X

i=0

µ⁽ⁱ⁾◦φ^⊗i+1◦∆⁽ⁱ⁾_C ⊗µ^(n−2−i)◦φ^⊗n−1−i◦∆⁽ⁿ_C⁻²⁻ⁱ⁾◦∆_C

+

n−1

X

i=0

µ⁽ⁱ⁻¹⁾◦φ^⊗i◦∆⁽ⁱ_C⁻¹⁾⊗µ^(n−1−i)◦φ^⊗n−i◦∆⁽ⁿ_C⁻¹⁻ⁱ⁾◦∆_C

−

n−2

X

i=0

µ⁽ⁱ⁾◦φ^⊗i+1◦∆⁽ⁱ⁾_C ⊗µ^(n−2−i)◦φ^⊗n−1−i◦∆⁽ⁿ_C⁻²⁻ⁱ⁾◦∆_C

!

=X

n∈N n

X

i=0

µ⁽ⁱ⁻¹⁾◦φ^⊗i◦∆⁽ⁱ_C⁻¹⁾⊗µ^(n−1−i)◦φ^⊗n−1◦∆⁽ⁿ_C⁻¹⁻ⁱ⁾

!

◦∆_C

= X

n∈N n

X

i=0

φ^?i⊗φ^?n−i

!

◦∆_C = X

r∈N

φ^?r⊗X

s∈N

φ^?s

!

◦∆_C

= (φ⊗φ)◦∆_C, soφ=P

n∈Nφ^?nfits. To prove the uniqueness, denoteφ=P

n∈Nφ_n, with Imφ_n⊂V^⊗n. We have

(φ⊗φ)◦∆_C =∆◦φ⇒µ◦(φ⊗φ)◦∆_C =µ◦∆◦φ.

Since (µ◦∆)(x₁· · ·x_k) = (k+1)x₁· · ·x_k, this last equation readsφ?φ= P

n∈N(n+ 1)φ_n. So for alln∈N, sinceφ₀=1ε_C,

n

X

i=0

φ_i? φ_n−i= (n+ 1)φ_n⇔(n−1)φ_n=

n−1

X

i=1

φ_i? φ_n−i. By induction, supposingφ_k =φ^?kfor 06k6n,nφ_n+1=P_n

i=1φ_i? φ_n+1−i =Pn

i=1φ^?n+1 =nφ^?n+1, so we also have φ_n+1 =φ^?n+1 and thus φ_n =φ^?n for all n ∈ N. So φ= P

n∈Nφ^?n is uniquely deter- mined byφ.

The coalgebra morphismφis said to be coinduced byφ:C⁺→ V and is computed as φ= P

n∈Nφ^?n, the geometric serie using the convolution with respect to the free multiplication µ and the co- multiplication∆_C.

Two coalgebra morphisms Φ,Ψ :TV ←C are equal if and only if there componentspr_V◦Φ andpr_V◦Ψ are equal. For example,µ_sh is coinduced bypr⊗ε+ε⊗pr, see equation (1.1.10).

Proposition 1.1.5 Letd :C⁺ →V be a linear map of degreej ∈Z. There exists a unique graded coderivation of degreejalongφ, notedd:C→ TV, i.e.∆◦d= (d⊗φ+φ⊗d)◦∆_C, such thatpr_V◦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)⊗1ε_C(x₂) +1ε_C(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+· · ·

As before, the composition withµ

∆◦d= (d⊗φ+φ⊗d)◦∆_C⇒µ◦∆◦d =d ? φ+φ ? d gives the uniqueness ofd.

Conversely, any graded coderivationD:C→ TV is determined by its componentd Bpr_V◦D. Moreover,d is determined by its re- strictionsd_k Bd|_V⊗k for any nonnegative integerk.

Taking the coalgebra C = (T V ,∆) and considering coderiva- tions alongφ=idTV, we note ford₁, d₂:TV →V

d₁◦_Gd₂Bd₁◦d₂=d₁◦(idTV? d₂? idTV) =d₁◦

∞

X

i,j=0

id^⊗i⊗d₂⊗id^⊗j : T V →V (1.1.7)

thegraded Gerstenhaber multiplication. So, the graded commutator [d₁, d₂] =d₁◦d₂−(−1)^|d1||d2|d₂◦d₁is a coderivation alongidTV which is coinduced bypr_V[d₁, d₂] =d₁◦_Gd₂−(−1)^|d1||d2|d₂◦d₁C[d₁, d₂]_G, thegraded Gerstenhaber bracketofd1andd2. There is then equality between

d◦_Gpr_V[d₁, d₂] =d◦[d₁, d₂] = (d◦_Gd₁)◦_Gd₂−(−1)^|d1||d2|(d◦_Gd₂)◦_Gd₁ and

d◦_G[d₁, d₂]_G=d◦_G(d₁◦_Gd₂)−(−1)^|d1||d2|d◦_G(d₂◦_Gd₁), known as the Gerstenhaber identity

(d◦_Gd₁)◦_Gd₂−(−1)^|d1||d2|(d◦_Gd₂)◦_Gd₁=d◦_G[d₁, d₂]_G. (1.1.8) So there is a graded pre-Lie identity for◦_G, so (Hom(T V , V),[, ]_G)

is a graded Lie algebra.

The structures◦_Gand [, ]_Gwere first defined by Gerstenhaber in [Ger63] forV[1] where the graded space wasV =V⁰.

Lemma 1.1.6 Letµ:V⊗V →V be a graded associative multiplication (of degree0).

Then the shifted mapd=µ[1] :V[1]⊗V[1]→V[1]is of degree1, and the associativity ofµis equivalent tod◦_Gd = 0.

Proof. We have that d :V[1]^⊗2→V[1]

d(a⊗b) =s⁻¹◦µ◦(s⊗s)(a⊗b) = (−1)^|a|s⁻¹µ(s(a)⊗s(b)) = (−1)^|a|ab,

and fora, b, c∈V[1],

(d◦_Gd)(a⊗b⊗c) = (µ[1]◦µ[1])(a⊗b⊗c)

=µ[1](µ[1]⊗id+id⊗µ[1])(a⊗b⊗c)

=µ[1](µ[1](a⊗b)⊗c) + (−1)^|a|µ[1](a⊗µ[1](b⊗c))

= (−1)^|a|µ[1](ab⊗c) + (−1)^|a|+|b|µ[1](a⊗bc)

= (−1)^|a|+|ab|(ab)c+ (−1)^2|a|+|b|a(bc) since|ab|=|s⁻¹(sa sb)|=|a|+|b|+ 1

= (−1)^2|a|+|b| a(bc)−(ab)c

= (−1)^|b|−1as_µ[1](a, b, c)

= 0.

1.1.4 Symmetric bialgebra

Thegraded symmetric bialgebraonV,SV =⊕_n∈

NSⁿV is defined as the quotient of the free algebraTV by the two-sided ideal gener- ated by all elements xy−(−1)^|x||y|yx in TV withx ∈V^|x|, y ∈V^|y|. The resulting associative multiplication, theshuffle multiplication

• is graded commutative,i.e. for two homogenous elementsa, b∈ SV we havea•b= (−1)^|a||b|b•a, and has for unit element1. So the quotientSV is a graded commutative associative algebra.

Moreover, the first comultiplication ∆_sh factors through the quotient and define onSV a graded cocommutative comultiplica- tion, also noted∆_sh. The spaceSV is then a graded commutative cocommutative bialgebra. It is the free graded commutative alge- bra generated byV.

For an integer n, a permutation σ from the symmetric group Sn

and ξ = x₁· · ·x_n ∈ V^⊗n, denote ξ^σ = x_σ(1)· · ·x_σ(n) the usual right action from S_n onT V. Defining the graded signature fromσ with respect toξas

e(x₁· · ·x_n, σ)B Y

16i<j6n

σ(i) + (−1)^{|xσ(i)||xσ(j)|}σ(j) i+ (−1)^|xi||xj|j

= Y

i<jandσ(i)>σ(j)

(−1)^{|xσ(i)||xσ(j)|},

there is a graded right action ξ^.σ =e(ξ, σ)ξ^σ fromSn onV^⊗n, be- causee(ξ, σ τ) =e(ξ, σ)e(ξ^σ, τ).

Using this action, we can give an explicit formula of the shuf- fle multiplication. Note Sh(k, n−k) the set of shuffle permuta- tions, i.e. permutations σ ∈ S_n such that σ(1) < · · · < σ(k) and σ(k+ 1)<· · ·< σ(n). Then