Formality theorems: from associators to a global formulation

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BLAISE PASCAL

Gilles Halbout

Formality theorems: from associators to a global formulation

Volume 13, n^o2 (2006), p. 313-348.

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Formality theorems: from associators to a global formulation

Gilles Halbout

Abstract

LetM be a differential manifold. LetΦbe a Drinfeld associator. In this paper we explain how to construct a global formality morphism starting fromΦ. More precisely, following Tamarkin’s proof, we construct a Lie homomorphism “up to homotopy" between the Lie algebra of Hochschild cochains onC^∞(M)and its cohomology(Γ(M,ΛT M),[−,−]S). This paper is an extended version of a course given 8 - 12 March 2004 on Tamarkin’s works. The reader will find explicit examples, recollections onG∞-structures, explanation of the Etingof-Kazhdan quantization- dequantization theorem, of Tamarkin’s cohomological obstruction and of globalization process needed to get the formality theorem. Finally, we prove here that Tamarkin’s formality maps can be globalized.

1. Introduction

Let M be a differential manifold andA=C^∞(M) the algebra of smooth differential functions over M. Formality theorems link commutative ob- jects with their non commutative analogs. More precisely, one has two graded Lie algebra structures:

- The space T_poly = Γ(M,ΛT M) of multivector fields on M. It is endowed with a graded Lie bracket[−,−]_S called the Schouten bracket (see [20]), extending the Lie bracket of vector fields (see Example 2.3 in section 1).

- The space D_poly = C(A, A) =^L_k≥0C^k(A, A), of regular Hochschild cochains (generated by differentialk-linear maps fromA^ktoAand support preserving). This vector space D_poly is also endowed with a differential graded Lie algebra structure given by the Gerstenhaber bracket[−,−]_G[9]

and coHochschild differentialb (see Example 2.4 in section 1).

We have:

Theorem 1.1. The cohomology H^∗(D_poly, b) of D_poly with respect tob is isomorphic to the space Tpoly ([15]).

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More precisely, one can construct a quasi-isomorphism of complexes ϕ¹ : (T_poly,0)→(D_poly, b),

called the Hochschild-Kostant-Rosenberg quasi-isomorphism ([15]); it is defined, forα ∈T_poly,f₁, . . . , f_n∈A, by

ϕ¹ :α7→ (f1, . . . , fn)7→ hα, df₁∧ · · · ∧dfni.

This map ϕ¹ is not a differential Lie algebra morphism but it is “up to (higher) homotopy”. Formality maps are the collection of those homotopies: they are maps, ϕ^1,...,1:ΛⁿT_poly→D_poly, forn≥0, such that

(d¹_T +d^1,1_T )◦ϕ=ϕ◦d^1,1_T , (1.1) where we have “extended” the Lie bracket [−,−]_S to a coderivationd^1,1_T : Λ^·T_poly → Λ^·T_poly, the Lie bracket [−,−]_G and the differential b to coderivations d^1,1_D and d¹_D: Λ^·D_poly → Λ^·D_poly and the maps ϕ^1,...,1 to morphisms of coalgebras :Λ^·Tpoly →Λ^·Dpoly on the corresponding cofree cocommutative coalgebras. In the first section of this paper we will recall precise definitions ofΛ^·E forEa graded vector spaces, of the above maps and of their “extension”.

Existence of such homotopies was proven for M = R^d by Kontsevich (see [18] and [19]) and Tamarkin (see [22]). They use different methods in their proofs. Kontsevich proved also that those maps can be globalized on a general manifold. When M is a Poisson manifold equipped with a Poisson bracket corresponding to a Poisson 2-tensor field π (such that [π, π]_S= 0), one can deduce the existence of a star-productm_? onM,i.e.

an associative product on A[[~]] for~a formal parameter:

m_? =m+~ϕ¹(π) +^X

n≥2

~ⁿ

n!ϕⁿ(πΛ· · ·Λπ).

Notice that until the end of the paper, we will use the notation Λ for the product on the exterior algebra Λ^·E and ∧ for the exterior product on T_poly.

The fact thatm_?is associative,i.e.[m_?, m_?]_G = 0, follows from equation (1.1) and one has ϕ¹(π) ={−,−}, the Poisson bracket.

We will follow Tamarkin’s proof and show how to build such homotopies. In the first three sections, we will suppose that M =R^d.

The paper is organized as follows:

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• In Section 2, we will make precise definitions of L∞ and G∞- structures and morphisms used to define the formality maps. Ex- plicit formulas will be given.

• In Section 3, we will show that the space D_poly can be endowed with aG∞-structure. This is where associators and Etingof-Kazhdan theorem will be needed. We will outline proofs by Etingof and Kazhdan and also by Enriquez.

• In Section 4, we will construct the formality maps when the manifold M = R^d. To do so, we will describe obstructions to such a construction and show that they vanish when M =R^d.

• In Section 5, we will prove that those maps can be globalized when M is an arbitrary manifold. To do so, we will follow Dolgushev’s approach ([2]) where the globalization process was done to local Kontsevich’s maps.

I would like to thank D. Manchon and D. Arnal for their invitation in Dĳon, D. Calaque, V. Dolgushev, G. Ginot and B. Keller for many useful suggestions and B. Enriquez and P. Etingof for helping better understand the Etingof-Kazhdan dequantization theorem.

2. G_∞-structures

The first aim of this section is to give a precise meaning to Equation (1.1) and to explain what we mean by “canonical extension” on ΛTpoly

or ΛD_poly. To do so, let us reformulate the definition of a Lie algebra and more generally of a L∞-algebra. For a graded vector space E, let us denote T E = T(E[1]) the free tensor algebra of E which, equipped with the coshuffle coproduct, is a bialgebra. The coshuffle coproduct ∆is defined on the generatorsx ofT E by∆(x) =x⊗1 + 1⊗x. Let us denote ΛE = S(E[1]) the free graded commutative algebra generated by E[1], seen as a quotient of T E. The coshuffle coproduct is still well defined on ΛE which becomes a cofree cocommutative coalgebra. One can write an explicit formula for the coproduct ∆ : ΛE →(ΛE)Λ(ΛE),

∆(γ₁Λ· · ·Λγ_n) = 1 2

n−1

X

k=1

X

ε∈Sn

sgn(ε)(γ_ε(1)Λ· · ·Λγ_ε(k))Λ(γ_ε(k+1)Λ· · ·Λγ_ε(n)),

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where sgn(ε) corresponds to the graded signature of the permutation defined, for any permutation ε of {1, ..., n} and any graded variables γ1, . . . , γn inE (with degree shifted by minus one), by the identity

γ₁· · ·γ_n=sgn(ε)γ_ε⁻¹₍₁₎· · ·γ_σ⁻¹_(n)

which holds in the free graded commutative algebra generated byγ1, . . . , γn. For E₁, E₂ ∈ E, E₁ΛE₂ will stand for the corresponding quotient of E1[1]⊗E2[1] in ΛE. We will use the notations TⁿE and ΛⁿE for the elements of degree n. We have now

Definition 2.1. A vector space E is endowed with a L∞-algebra (Lie algebra “up to homotopy”) structure if there are degree one linear maps d^1,...,1:Λ^kE →E[1] such that the asociated coderivations (extended with respect to the cofree cocommutative structure on ΛE) d: ΛE → ΛE, satisfy d◦d= 0 wheredis the coderivation

d=d¹+d^1,1+· · ·+d^1,...,1+· · ·.

One can again write explicit formulas for the extensions of the maps as coderivations (∆◦d^1,...,1= (d^1,...,1⊗Id+Id⊗d^1,...,1)◦∆):

d(γ1Λ· · ·Λγn) =d^1,...,1(γ1Λ· · ·Λγn) +

n−1

X

k=1

X

ε∈Sn

sgn(ε)d^1,...,1(γ_ε(1)Λ· · ·Λγ_ε(k))Λγ_ε(k+1)Λ· · ·Λγ_ε(n).

In particular, we have

Remark 2.2. A differential Lie algebra (E, d,[−,−])is aL∞-algebra with structure maps d¹ =d[1], d^1,1 = [−,−][1] and d^1,...,1:Λ^kE → E[1] are 0 for k≥3.

Let us recall the two examples T_poly and D_poly:

Example 2.3. The space T_poly is a graded Lie algebra (and so a L∞- algebra) with0differential and Schouten bracket[−,−]_Sdefined as follows [α, β∧γ]S = [α, β]S∧γ+ (−1)^|α|(|β|+1)β∧[α, γ]S (2.1) forα, β, γ∈T_poly. Forf ∈Γ(M,Λ⁰T M) =C^∞(M)andα∈Γ(M,Λ¹T M) we set [α, f]S = α·f, the action of the vector field α on f. The grading

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on T_poly is defined by |α| = n ⇔ α ∈ Γ(M,Λⁿ⁺¹T M) and the exterior product is graded commutative:

∀α, β∈Γ(M,ΛT M), α∧β = (−1)(|α|+1)(|β|+1)

β∧α.

Let us denote dT the associated coderivation (d^1,1_T is corresponding to [−,−]_S[1]). One can check that the Jacobi identity for[−,−]_Sis equivalent tod^1,1_T ◦d^1,1_T = 0.

Example 2.4. Similarly, D_poly is a differential graded Lie algebra (and so a L∞-algebra). Its bracket, the Gerstenhaber bracket [−,−]_G, is defined, for D, E∈Dpoly, by

[D, E]_G={D|E} −(−1)^|E||D|{E|D}, where

{D|E}(x₁, . . ., xd+e−1) =^X

i≥0

(−1)^|E|·iD(x1, . . ., xi, E(xi+1, . . ., xi+e), . . .).

The space Dpoly has a grading defined by |D|= k ⇔ D ∈ C^k+1(A, A).

Finally, its differential is the coHochschild differential b= [m,−]_G, where m ∈C²(A, A) is the commutative multiplication onA. Let us denote dD

the associated coderivation (d^1,1_D corresponding to [−,−]_G[1] and d¹_D to b[1]). One can check that Jacobi identity for[−,−]_G,b² = 0and compatibility betweenband[−,−]_Gare equivalent to(d¹_D+d^1,1_D )◦(d¹_D+d^1,1_D ) = 0.

One can now define the generalization of Lie algebra morphisms:

Definition 2.5. A L∞-morphism between two L∞-algebras (E1, d1 = d¹₁ +· · ·) and (E₂, d₂ = d¹₂ +· · ·) is a morphism of differential cofree coalgebras, of degree 0,

ϕ: (ΛE₁, d₁)→(ΛE₂, d₂).

In particularϕ◦d1 =d2◦ϕ. Asϕis a morphism of cofree cocommutative coalgebras (i.e. ∆₂ϕ = (ϕ⊗ϕ)∆₁ where ∆₁ and ∆₂ are the coproducts on E1 andE2),ϕis determined by its image on the cogenerators, i.e., by its components: ϕ^1,...,1 : Λ^kE1 →E2[1].Again one gets a general formula

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for ϕ:

ϕ(γ1Λ· · ·Λγn) =ϕ^1,...,1(γ1Λ· · ·Λγn) +

n−1

X

p=1

1 p!

k1+···+kp=n

X

k1,...,kp≥1

X

ε∈S_n

sgn(ε)ϕ^1,...,1(γ_ε(1)Λ· · ·Λγ_ε(k))Λ· · ·Λϕ^1,...,1(γ_ε(n−k+1)Λ· · ·Λγ_ε(n)) where the signs are Quillen’s signs corresponding to permutations of odd elements. Now equation (1.1) can be rewritten as follows: let dT and dD

correspond respectively to the Lie algebra structure on T_poly and to the differential Lie algebra structure on Dpoly. We want to construct a L∞- morphism ϕsuch thatϕ¹ is the Hochschild-Kostant-Rosenberg map and:

ϕ◦dT =dD◦ϕ.

If one tries to construct the maps ϕ^1,...,1 : ΛⁿTpoly → Dpoly by induction on n, one will find obstructions in the non acyclic Chevalley Eilenberg complex Hom(ΛT_poly, T_poly,[d_T,−]).

Tamarkin’s idea was then to extend the structure (or increase the con- strains) to reduce the obstructions. Indeed,T_polyhas a Gerstenhaber structure. It would be convenient to find such a structure on D_poly (we will see thatD_poly has actually aG∞-structurei.e.an “up to homotopy” Gersten- haber structure) and to construct a G∞-morphism between them (that restricts to a L∞-morphism on the corresponding Lie algebra structures).

Thanks to the addition of those extra operations, we will see that obstructions to the construction of G∞-morphisms will vanish in the case M =R^d. Let us end this section by some recollections on G∞-structures.

We will follow works of Ginot ([10]).

To define a G∞-structure on E, we will need a bigger space than ΛE. Let us denote^cT(E) the cofree tensor coalgebra of E (with coproduct ∆⁰). We will sometimes use the notation E^⊗·. Equipped with the shuffle product • (defined on the cogenerators ^cT(E)⊗^cT(E) → E as pr⊗ε+ε⊗pr, where pr : ^cT(E) → E is the projection and ε is the counit), it is a bialgebra. Let ^cT(E)⁺be the augmentation ideal. We note

cT(E) =^cT(E)⁺/(^cT(E)⁺•^cT(E)⁺)the quotient by the shuffles. It has a graded cofree coLie coalgebra structure (with coproduct δ = ∆⁰−∆^0op), see [12] for example. ThenS(^cT(E)[1])has a structure of cofree coGersten- haber algebra (i.e., equipped with cofree coLie and cofree cocommutative

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coproducts δ and ∆satisfying compatibility condition). One can write δ explicitly: for γ_i ∈E^⊗pⁱ,

δ(γ₁Λ· · ·Λγ_n) =

Xsgn(ε)s_kγ_ε(1)Λ· · ·Λγ_ε(i)Λ(α^k₁· · ·α^k_j)⊗(α_j+1^k · · ·α^k_p

k)Λγ_ε(i+1)Λ· · ·Λγ_ε(n), where the sum is over all integers1≤k≤n,1≤j≤p_k and all permuta- tionsεfixingkwhich are(i, n−1−i)-shuffles on{1, . . . , n}−{k}. We have denotedγ_k=α^k₁· · ·α^k_jα^k_j+1· · ·α^k_p_kand the signs_k = (−1)^(|α¹^|+···+|α^j^|)(p^k^−j). Moreover, we still have:

∆(γ₁Λ· · ·Λγ_n) =^Xsgn(ε)(γ_ε(1)Λ· · ·Λγ_ε(i))Λ(γ_ε(i+1)Λ· · ·Λγ_ε(n)), where the sum is over (i, n−i)-shuffles. We use the notation ^cT^m(E) for the elements of degree m, and, forγ₁¹, . . . , γ_n^pⁿ ∈E, we have

|γ₁¹⊗ · · · ⊗γ₁^p¹Λ· · ·Λγ_n¹⊗ · · · ⊗γ_n^pⁿ|=

p1

X

i1

|γ₁ⁱ¹|+· · ·+

pn

X

in

|γ_nⁱⁿ| −n.

Definition 2.6. A vector space E is endowed with aG∞-algebra (Ger- stenhaber algebra “up to homotopy”) structure if there are degree one linear maps d^p¹^,...,p^k: ^cT^p¹(E)Λ· · ·Λ^cT^p^k(E) ⊂ Λ^{k c}T E → E[1] such that the associated coderivations (extended with respect to the cofree coGer- stenhaber structure on Λ^cT(E)) d:Λ^cT(E) →Λ^cT(E) satisfies d◦d= 0 where dis the coderivation

d=d¹+d^1,1+· · ·+d^p¹^,...,p^k +· · · . More details onG∞-structures are given in [10].

In particular we have

Remark 2.7. If(E, d,[−,−],∧)is a differential Gerstenhaber algebra, then E[1] is a G∞-algebra with structure maps d¹ = d[1], d^1,1 = [−,−][1], d² =∧[1] and otherd^p¹^,...,p^k:^cT^p¹(E[1]) Λ· · ·Λ^cT^p^k(E[1])→E[2]are 0.

Applying this remark to the spaces T_poly andD_poly we get

Example 2.8. The space T_poly is a graded Gerstenhaber algebra and so a G∞-algebra with maps d^1,1_T = [−,−]_S[1] and d²_T = ∧[1] the exterior product. It is clear thatd²_T is a well defined mapT_poly^⊗2 →T_poly (because it is graded commutative).

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Let us, as an exercise, extend maps d^1,1_T and d²_T for degree 0 elements α, β, γ inT_poly.

d²_T(α⊗β⊗γ) =d²_T(α⊗β)⊗γ+α⊗d²_T(β⊗γ) = (α∧β)⊗γ−α⊗(β∧γ).

and so the conditiond²_T ◦d²_T(α⊗β⊗γ) = (α∧β)∧γ−α∧(β∧γ) = 0is equivalent to the associativity of the map d²_T.

In the same way, we have:

(d^1,1_T ◦d²_T +d²_T ◦d^1,1_T ((α⊗β)∧γ)

= [α∧β, γ]S+d²_T(α⊗d^1,1_T (β∧γ))−d²_T(d^1,1_T (γ∧α)⊗β)

= [α∧β, γ]_S+α∧[β, γ]_S−[γ, α]_S∧β = 0,

by compatibility between [−,−]_S and ∧. So all the identities defining the Gerstenhaber algebra structure on T_poly can be summarized into the unique relation (d^1,1_T +d²_T)◦(d^1,1_T +d²_T) = 0.

Example 2.9. The space Dpoly is not a (graded) Gerstenhaber algebra when equiped with the product of cochains ∪ defined, for D, E ∈ D_poly and x₁, . . . , x_|D|+|E|+2 ∈A, by

(D∪E)(x₁, . . ., x|D|+|E|+2) = (−1)^γD(x1, . . ., x|D|+1)E(x|D|+2, . . ., x|D|+|E|+2) where γ = (|E|+ 1)(|D|+ 1). The projection of this product on the cohomology of (D_poly, b) is the exterior product ∧, but unfortunately (Dpoly,[−,−]_G, ∪, b) is not a Gerstenhaber algebra: one can see, for example, that ∪ is not a graded commutative product and thus can not be defined as a map Dpoly⊗2 → Dpoly. More generally, Gerstenhaber’s cachain structure have the same “failure”, only the cohomology behaves well.

We will show in Section 2 that it can be equiped with aG∞-structure.

One can now define the generalization of Gerstenhaber morphisms:

Definition 2.10. A G∞-morphism between two G∞-algebras (E1, d1 = d¹₁+d²₁ +· · ·) and (E₂, d₂ =d¹₂+d²₂+· · ·) is a morphism of differential coGerstenhaber coalgebras, of degree 0,

ϕ: (Λ^cT(E₁), d₁)→(Λ^cT(E₂), d₂).

In particularϕ◦d₁ =d₂◦ϕ. Asϕis a morphism of cofree coGerstenhaber coalgebras, ϕ is determined by its image on the cogenerators, i.e., by its

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components: ϕ^p¹^,...,p^k:^cT^p¹(E1)Λ· · · Λ^cT^p^k(E1)→ E2[1].As an example, for degree0 elements α, β, γ inE1, one has

ϕ((α⊗β)Λγ) =ϕ^2,1(α⊗β, γ)

+ϕ¹(α)⊗ϕ^1,1(βΛγ)−ϕ^1,1(γΛα)⊗ϕ¹(β) +ϕ²(α⊗β)Λϕ¹(γ)

+ (ϕ¹(α)⊗ϕ¹(β))Λϕ¹(γ).

3. A G∞-structure on the space of cochains

The objective of this section is to prove the following proposition ([22]).

Proposition 3.1. There exists aG∞-structure onD_polygiven by a coderiva- tion dD such that if dD = ^P

l≥1, p₁+···p_n=l

d^p_D¹^,...,pⁿ, then dD◦dD = 0 and (1) d¹_D is the Hochschild differential b.

(2) d^1,1_D is the Gerstenhaber bracket [−,−]_G.

(3) d²_D is the cup product ∪, up to a Hochschild coboundary.

(4) d^p_D¹^,...,pⁿ = 0 for n >2.

3.1. Construction of the G_∞-structure

We first reformulate this problem: letLD =⊕Dpoly⊗nbe the cofree coLie coalgebra on D_poly (see Section 2 for the notation). Since L_D is a cofree coLie coalgebra, a differential Lie bialgebra structure on LD is uniquely determined by the restriction to cogenerators of the Lie bracket and the differential (which are coderivations on LD) and so by degree one maps l_Dⁿ: D_poly^⊗n → D_poly (for the differential L_D → L_D), and maps l^p_D¹^,p²: D_poly^⊗p¹ΛD_poly^⊗p² → D_poly (for the Lie bracket L_DΛL_D → L_D). The following lemma is well known.

Lemma 3.2. Suppose we have a differential Lie bialgebra structure on the coLie coalgebra L_D, with differential and Lie bracket respectively de- termined by maps lⁿ_D and l^p_D¹^,p² as above. Then Dpoly has aG∞-structure

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given, for all p, q, n≥1, by

dⁿ_D =lⁿ_D, d^p,q_D =l^p,q_D and d^p_D¹^,...,p^r = 0 for r ≥3.

Proof. The map dD = ^P_i≥0lⁱ_D +^P_p₁_,p₂_≥0l^p_D¹^,p² : Λ^·LD → Λ^·LD is the Chevalley-Eilenberg differential on the differential Lie algebra L_D; it sat-

isfies dD◦dD= 0.

Thus to obtain the desired G∞-structure on D_poly, it is enough to define a differential Lie bialgebra structure on L_D given by maps lⁿ_D and l^p_D¹^,p² with l¹_D =b,l^1,1_D = [−,−]_G and l²_D =∪“up to homotopy”.

Let us now give an equivalent formulation of our problem, which is stated in terms of the associated operads in [22]:

Proposition 3.3. Suppose we have a differential bialgebra structure on the cofree tensorial coalgebra T_D = ⊕_n≥0 D_poly^⊗n with differential and multiplication given respectively by maps aⁿ_D:D_poly^⊗n→D_poly anda^p_D¹^,p²: D_poly^⊗p¹⊗ D_poly^⊗p² → D_poly. Then we have a differential Lie bialgebra structure on the coLie coalgebraL_D =⊕n≥0 D_poly^⊗n, with differential and Lie bracket respectively determined by maps lⁿ_D and l^p_D¹^,p² where l¹_D =a¹_D, l_D^1,1 is the anti-symmetrization of a^1,1_D and l²_D =a²_D “up to homotopy”.

A differential bialgebra structure on the cofree tensorial coalgebra ⊕V^⊗n associated to a vector spaceV is often called aB∞-structure onV, see [1].

Proof. The proof relies on the existence of a quantization/dequantization functor, that we will recall in the next subsection. Let V be a finite- dimensional vector space andV^∗be the dual space. A differential bialgebra structure on the cofree coalgebra^cT V =⊕_n≥0V^⊗nis defined on the cogenerators by maps aⁿ: V^⊗n →V (n≥2), corresponding to the differential P

n≥0aⁿ:^cT V → ^cT V, and maps a^p¹^,p²:V^⊗p¹ ⊗V^⊗p² → V (p1, p2 ≥0), corresponding to the product^P_p₁_,p₂_≥0a^p¹^,p²:^cT V ⊗^cT V →^cT V. We can define dual maps of those maps to get again a differential bialgebra with differential D:Tˆ→Tˆ and coproduct ∆:Tˆ→ Tˆ⊗ˆTˆ, where Tˆ is the completion of the tensor algebra ⊕_n≥0V^∗⊗n. The differential and coproduct Dand ∆are defined now on the generators of the free algebraTˆ by maps a^n∗: V^∗ → V^∗⊗n and a^p¹^,p²^∗: V^∗ → V^∗⊗p¹ ⊗V^∗⊗p². The tensor algebra

⊕n≥0V^∗⊗n is graded as follows: |x|=p whenx∈V^∗⊗p.

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Similarly, if we consider a differential Lie bialgebra structure on the cofree coLie coalgebraL=⊕_n≥0V^⊗n, the dual mapsdandδof the structure maps ^P_n≥0lⁿ and ^P_p₁_,p₂_≥0l^p¹^,p² induce a differential Lie bialgebra structure onL, the completion of the free Lie algebraˆ ⊕_n≥0Lie(V^∗)(n)on V^∗, whereLie(V^∗)(n)is the subspace of element of degree n.

We now replace formally each element x of degree nin Tˆ (resp. L) byˆ hⁿx, where h is a formal parameter. Letting|h|=−1, we easily see that it is equivalent to define

• a differential associative (respectively Lie) bialgebra structure on the associative (resp. Lie) algebras (⊕_n≥0V^∗⊗n)[[h]]

(resp.(⊕_n≥0Lie(V^∗)(n))[[h]]) with the product and coproduct be- ing of degree zero

• or a differential associative (resp. Lie) bialgebra structure on the associative (resp. Lie) algebraTˆ (resp.L).ˆ

Note that those two bullets are dual. Thus we have a differential free coalgebra ( ˆT[[h]], D,∆).

We can apply now Etingof-Kazhdan’s dequantization theorem for graded differential bialgebras ([7] and Appendix in [11]) to our particular case where we start from a differential bialgebra free as an algebra ( ˆT ,∆, D):

this proves that

Proposition 3.4. There exists a Lie bialgebra ( ˆL,[−,−], δ, d), generated as a Lie algebra byV^∗and an injective mapIEK:L[[h]]ˆ →(⊕n≥0V^∗⊗n)[[h]]

such that

(1) the restriction I_EK: V^∗→V^∗ is the identity,

(2) the maps I_EK, δ and [−,−] are given by universal formulas (i.e.

depending only on ∆ and the product of Tˆ),

(3) I_EK([a, b]) = I_EK(a)I_EK(b)−I_EK(b)I_EK(a) +O(h), for all a, b ∈ L[[h]],ˆ

(4) (∆−∆^op)I_EK=hI_EKδ+O(h²), (5) IEK◦d=D◦IEK

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(6) if we apply Etingof-Kazhdan’s quantization functor (see [6]) to the Lie bialgebra (⊕_n≥0Lie(V^∗)ⁿ[[h]], δ) we get the bialgebra ((⊕_n≥0V^∗⊗n)[[h]],∆) back.

The last condition implies that Lˆ is free as a Lie algebra because Tˆ is free as an algebra. Moreover the structure maps l^p∗_D and l^p,q∗_D onLˆ satisfy l_D^1∗ = a^1∗_D, l^1,1∗_D is the anti-symmetrization of a^1,1∗_D ans l^2∗_D = a^2∗_D “up to homotopy”. Taking now dual maps, we get the result.

Remark 3.5. Here one strongly used the quantization/dequantization theorem. Indeed, if one only takes the anti-symmetrization and the classical limit to get the wanted Lie algebra structure on L_D, one will lose the information on degree 2 maps and in particular the information on l_D². Recall that we wanted l_D² =∪“up to homotopy” and by taking the naive classical limit one would get l²_D = 0 which will then only give the Lie algebra structure on D_poly that we started with !

By Proposition 3.3, the problem of defining a differential Lie bialgebra structure on LD given by maps l_Dⁿ and l_D^p¹^,p² with l¹_D =b, l^1,1_D = [−,−]_G and l²_D = ∪ “up to homotopy” is now equivalent to defining a differential bialgebra structure on TD given by maps aⁿ_D: Dpoly⊗n → Dpoly

and a^p_D¹^,p²: D_poly^⊗p¹⊗D_poly^⊗p² → D_poly where a¹_D = b, a^1,1_D is the product {−|−} defined in Section 0 and a²_D = ∪ “up to homotopy”. Indeed, the anti-symmetrization of {−|−} is by definition[−,−]_G. The latter can be achieved using the braces operations (defined in [9]) acting on the Hochschild cochain complex D_poly = C(A, A) for any algebra A. The braces operations are mapsa^1,p_D :D_poly⊗D_poly^⊗p →D_poly(p≥1) defined, for all homogeneous D, E₁, . . . , E_p ∈D_poly^⊗p+1 and x₁, . . . , x_d∈A (with d=|D|+|E₁|+· · ·+|E_p|+ 1), by

a^1,p_D (D⊗(E1⊗ · · · ⊗Ep))(x1⊗ · · · ⊗xd) =

X(−1)^τD(x1, . . ., xi1, E1(xi1+1, . . .), . . ., Ep(xip+1, . . .), . . .) where τ =^P^p_k=1i_k(|E_k|+ 1). It is clear that a^1,1_D corresponds to the map {−,−}. Now Theorem 3.1 in [23] asserts (see also [9] and [17]) that:

• The maps a^1,p_D : D_poly⊗D_poly^⊗p → D_poly, a^q≥2,p_D = 0 and the degree 0 shuffle product determine a coderivation ? = ^Pa^p,q_D on

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the cofree tensorial coalgebraT_D =⊕_n≥0D_poly^⊗nwhich turnsT_D into a bialgebra.

• Similarly takinga¹_D to be the Hochschild coboundaryband a²_D to be the cup-product ∪, and a^q≥3_D = 0, the coderivation d= ^Paⁿ_D defines a differential structure on the tensor coalgebraT_D.

• These maps yield a differential bialgebra structure (T_D, ?, d) on the cofree coalgebra TD.

Actually, one only need to prove the associativity condition as the differential is given by the commutator (with respect to the product ?) [m,−]

with the multiplicationm onA. Let us prove the three points for the first orders with respect to the degree:

• Let us check thata¹_D+a²_D is a differential. ForA, B inD_poly one gets:

(a¹_D +a²_D)◦(a¹_D+a²_D)(AB) = (b+∪)(bAB±AbB+A∪B)

=b(A∪B) +bA∪B±A∪bB = 0.

• Let us check the associativity of ?=a^1,1_D +a^1,2_D +· · · up to order 2. For A, B, C inDpoly, one gets (here we forget the signs):

(A ? B)? C= (AB+BA+{A, B})? C

={A, C}B+A{B, C}+{B, C}A+B{A, C}

+{A, B}C+C{A, B}+{{A, B}, C}

+ABC (and other permutations inABC) A ?(B ? C) =A ?(BC+CB+{B, C})

={A, B}C+B{A, C}+{A, C}B+C{A, B}+a^1,2_D (A, BC) +a^1,2_D (A, CB) +A{B, C}+{B, C}A+{A,{B, C}}

+ABC (and other permutations inABC), and the result follows from

{{A, B}, C}=a^1,2_D (A, BC) +a^1,2_D (A, CB) +{A,{B, C}}.

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• Let us check the compatibility condition between?=a^1,1_D +a^1,2_D +

· · · and the differential d= a¹_D +a²_D up to order 2. For A, B in D_poly, one gets (here again we forget the signs):

d(A ? B) = (b+∪)(AB+BA+{A, B})

=bAB+AbB+bBA+BbA+A∪B+B∪A+b{A, B}, dA ? B+A ? dB=bAB+BbA+AbB+bBA+{bA, B}+{A, bB},

and the result follows from

A∪B+B∪A={bA, B}+{A, bB} −b{A, B}.

Using this result, we can successively apply Proposition 3.3 and Lemma 3.2 to obtain the desired G∞-structure on D_poly given by maps d^p_D¹^,...,p^k such that d¹_D =b,d^1,1_D = [−,−]_G and d²_D =∪ “up to homotopy” (i.e., up to a coboundary). Moreover, one remembers that maps d^p_D¹^,...,p^k are 0 for k >2.

3.2. The quantization/dequantization fonctor Let us recall the definition of a Drinfeld associator (cf[4]):

Let T_n be the algebra generated by elements t_ij, 1 ≤ i, j ≤ n, i 6= j, with defining relations tij = tji, [tij, tlm] = 0 for i, j, l, m distincts and [tij, t_ik +t_jk] = 0 for i, j, k distincts. Let P1, . . . , Pn be disjoint subsets of {1, . . . , m}. There exists a unique homomorphism ρ_P₁_,...,P_n: T_n → T_m defined by

ρ_P₁_,...,P_n(t_ij) = ^X

p∈P_i,q∈P_j

t_pq.

For any X ∈ Tn, we denote ρP1,...,Pn(X) by X^P¹^,...,Pⁿ. Let Φ ∈ T3. The relation

Φ^1,2,34Φ^12,3,4= Φ^2,3,4Φ^1,23,4Φ^1,2,3

in T4[[~]] is called the pentagon relation. Let B = e^~^t¹²^/2 ∈ T2[[~]]. The relations

B^12,3 = Φ^3,1,2B^1,3(Φ^1,3,2)⁻¹B^2,3Φ^1,2,3, B^1,23= (Φ^2,3,1)⁻¹B^1,3Φ^2,1,3B^1,2(Φ^1,2,3)⁻¹ in T3[[~]] are called the hexagon relations.

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An element Φ ∈ T3 satisfying the pentagon and hexagon relations is called a Drinfeld associator. Such associators exist over C([3]). They are obtained from the KZ equations. Drinfeld also prove that such associators exist over Q.

In this subsection we recall the following theorem (see appendix in [11]) which gives, as a consequence, Proposition 3.4 (here is where an associator Φ is used):

Theorem 3.6. There exists an equivalence of categories DQΦ: DGQUE→DGLBA_~

from the category of differential graded quantized universal enveloping graded algebras to that of differential graded Lie graded bialgebras such that ifU ∈Ob(DGQUE)anda=DQ_Φ(U), thenU/~U =U(a/~a), where U is the universal algebra functor, taking a differential graded Lie graded algebra to a differential graded graded Hopf algebra.

This theorem is a consequence of the Etingof-Kazhdan quantization theorems. The key point is that the quantization theorem is “universal” and so will be valid for any symmetric category and so for complexes (V^·, d^·).

A right way to understand the “universality” is to use the language of operads and props. We will not recall the definitions in this paper.

Let us outline the construction of the quantization functor starting with an associator Φ. Let(g, δ) be a Lie bialgebra. Let D=g⊕g^∗ be its associated Lie bialgebra double. Letr ∈g⊗g^∗∈D^⊗2 be the canonicalr- matrix (corresponding to the identity map) andt=r+r^2,1∈S²(D)^D. Let us consider the homomorphism Tn → U(D)^⊗n sending tij to t^i,j (where components of tare put in thei-th and j-th place in the tensor product).

We will still denote by Φthe image ofΦ by this homomorphism.

We will use the standard notation for the coproduct-insertion maps:

we say that an ordered set is a pair of a finite set S and a bĳection {1, . . . ,|S|} → S. For I1, . . . , Im disjoint ordered subsets of {1, . . . , n}, (U,∆) a Hopf algebra anda∈U^⊗m, we define

a^I¹^,...,Iⁿ=σ_I₁_,...,I_m◦(∆^(|I¹^|)⊗ · · · ⊗∆^(|Iⁿ^|))(a),

with ∆⁽¹⁾ = Id, ∆⁽²⁾ = ∆, ∆⁽ⁿ⁺¹⁾ = (Id^⊗n−1 ⊗ ∆) ◦ ∆⁽ⁿ⁾, andσ_I₁_,...,I_m :U^⊗^Pⁱ^|Iⁱ^|→U^⊗nis the morphism corresponding to the map {1, . . . ,^P_i|I_i|}→ {1, . . . , n} taking (1, . . . ,|I₁|) toI1,(|I₁|+ 1, . . . ,|I₁|+

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|I₂|)toI2, etc. WhenU is cocommutative, this definition depends only on the sets underlying I₁, . . . , I_m.

We get that (U(D)[[~]], m₀,∆₀, R₀ = e^~^t/2,Φ) is a quasi-triangular quasi-Hopf algebra ([4]). Quasi-triangular means that

∆²¹₀ (a) =R∆0(a)R⁻¹

for all a ∈U(D) and quasi-Hopf means that the coproduct ∆₀ is quasi- coassociative, that is to say

(Id⊗∆₀)(∆0(a)) = Φ(∆0⊗Id)∆0(a)Φ⁻¹

for all a ∈ U(D). To make this quasi-Hopf algebra into a Hopf algebra, one has to twist Φ into the identity, that is to say one has to construct J ∈U(D)^⊗2 such that

((1⊗J)((Id⊗∆₀)(J)))⁻¹(J⊗1)(∆₀⊗Id)(J) = (J^2,3J^1,23)⁻¹J^1,2J^12,3 = Φ.

(3.1) Then (U(D)[[~]], m₀,Ad(J)◦∆₀, R=J^2,1e^~^t/2J⁻¹) is a Hopf algebra.

Suppose now we have constructed such a J (actually, we ask J to have also good “polarization” properties), setH ={(ξ⊗Id)R, ξ∈U(D)^∗[[~]]}.

It is a Hopf subalgebra ofU(D)[[~]]. LetU_~(g)be~-adic completion. More precisely, let I be the maximal ideal of H,U_~(g) is the ~-adic completion of the subalgebra ^P_n≥0~⁻ⁿIⁿ in H⊗_k[[~]]k((~)). It is clear that U_~(g)is isomorphic to U(g)[[~]] and so (U_~(g),Ad(J)◦∆₀) is then a quantization of (g, δ). Notice that the product in U_~(g) is not the same as the one in U_~(D)(and so the product in U(D)) as the algebra isomorphism U_~(g)' U(g)[[~]] is not the identity (which itself is not an algebra morphism).

Let us end this subsection showing how one can construct the twist J. In [6], the construction was done using the “categorical yoga” and one gets a general formula:

J = (φ⁻¹⊗φ⁻¹) (Φ^1,2,34)⁻¹φ^2,3,4se^~^t^2,3^/2(φ^2,3,4)⁻¹Φ^1,2,34(1+⊗1₊⊗1₋⊗1₋), whereM₊andM−are respectively the Verma module Ind^D_g 1and Ind^D_g^∗1, 1₊, and 1− are respectively the generators of those module over U(g^∗) and U(g) and φis the isomorphism U(D)→M+⊗M− generated by the assignment 1→1₊⊗1−. Finally,s is the twist in the tensor product. As an exercise, let us calculate the first terms of J. Let {a_i} be a basis ofg

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and {bⁱ} its dual basis, a basis of g^∗. So r =^Pai ⊗bⁱ. Let us write the structure constants:

[ai, b^j] =c^k_ija_k, δa_k=^Xf_k^ijaiΛaj

and so [bⁱ, b^j] =f_k^ijb^k and [ai, b^j] =f_i^jkak−c^j_ikb^k. Starting from an associator Φ = 1 + ^~₂₄²[t12, t23] +O(~³), one gets

J = 1 +~

2r+~² 1

4(a_ja_i⊗b^jbⁱ+f_i^jkcⁱ_jla_k⊗b^l)

−c^j_ik

12bⁱa_j⊗b^k−c^j_ik

24bⁱ⊗b^ka_j− f_i^jk

12 a_k⊗bⁱa_j−f_i^jk

24bⁱa_k⊗a_j.

To get universal formulas, one has then to reorder the terms in J. In [5], Enriquez proposed a cohomological construction of the twist J. He looks for this element in a “universal” algebra Uuniv made from the r-matrix. The definition is rather complicated and uses the language of props. We will retain that it is generated by the components of r, i.e.

words in {a_i} and {b^j} (with as many a’s as b’s) and the relations (the r-matrix relations):

aba⁰b⁰=aa⁰bb⁰+aa⁰[b⁰, b] + [a⁰, a]bb⁰.

This allows to write all the a’s on the left hand side and all theb’s in the right hand side. In the same spirit one can define

U_univ^⊗n =⊕_N≥0 ((F A_N)^⊗n)Σiδi⊗((F A_N)^⊗n)Σiδi

σN

where F A_N is the free algebra with generators x_i, i = 1, . . . , N, graded by ⊕_iNδi (xi has degree i). We view Φ as an element of U_univ^⊗3 and we will build J = 1 +~^r₂ +· · · ∈ U_univ^⊗2 such that equation (3.1) is fulfiled in U_univ^⊗3 . The construction is made by induction. Suppose we have built J = 1 +· · ·+~ⁿJn+· · · up to order n−1. Equation (3.1) at order nis equivalent to

d^coHo₂ Jn= Φn+hJ₁, . . . , Jn−1i

where Φ_n is the ~ⁿ component of Φ, hJ₁, . . . , Jn−1i is an expression in- volving only component Jk,k ≤n−1, and d^coHo_n :U_univ^⊗n →U_univ^⊗n+1 is the

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coHochschild differential:

d^coHo_n (j) =j12,3,...,n+1−j1,23,...,n+1+ (−1)ⁿ⁺¹j1,2,...,nn+1+j2,...,n,n+1

+ (−1)ⁿj^1,2,...,n. It is well known thatkerd^coHo_n =Imd^coHo_n−1 ⊕Λⁿ(D_univ)(this is true for any enveloping algebra). For any choice of J_k,k≤n−1,Φ_n+hJ₁, . . . , Jn−1i is in Kerd^coHo₂ . Moreover, one can always replaceJn−1 withJn−1+λn−1

(λn−1 ∈Λ²(D_univ)) so that we still have a solution up to ordern−1. The equation we want to solve now is the following equation with unknown (J_n, λn−1):

d^coHo₂ Jn= Φn+hJ₁, . . . , Jn−1+λn−1i=Cn+f(λn−1),

wheref:Λ²(D_univ)→U_univ^⊗3 ,λn−1 7→ hJ₁, . . . , Jn−1+λn−1i−hJ₁, . . . , Jn−1i, and d^coHo₃ C_n= 0, soC_n=d^coHo₂ K_n+µ_n, withµ_n∈Λ³(D_univ).

One hasd^coHo₃ (f(λn−1)) = 0sof(λn−1) =d^coHo₂ f⁰(λn−1) +Alt(f(λn−1)).

We get after computation Alt(f(λn−1)) = 1

6|[r, λ_n−1]|

( = [r^1,2, λ^1,3_n−1] + [r^1,2, λ^2,3_n−1] + [r^1,3, λ^2,3_n−1] + [λ^1,2_n−1, r^1,3] + [λ^1,2_n−1, r^2,3] + [λ^1,3_n−1, r^2,3]).

So one wants to solve

d^coHo₂ (Jn−f⁰(λn−1)−Kn) = 1

6|[r, λn−1]|+µn.

Actually, we have a complex, making|[r,−]|into a differential: when0≤ k≤n−1, let us define

(Id^⊗k⊗Id^⊗n−k−1)_univ : (D^⊗n)_univ →(D^⊗k⊗Λ(D)⊗D^⊗n−k−1)_univ by

a7→[r^k,k+1, a1,...,k,k+2,...,n+1+a1,...,k−1,k+1,...,n+1].

Then we have a complex ((Λ^·(D))univ, ∂^·) where

(Λ^k(D))_univ 3x7→∂^k(x) =Alt((∂⊗Id^⊗k−1)_univ(x))∈(Λ^k+1(D))_univ. It turns out that the 3-rd cohomology group of that complex is 0 if the

“degree” in a’s and b’s is greater than 3 and is spanned by the class of

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[t^1,2, t^2,3]otherwise. Moreover, one checks that Alt((∂⊗Id^⊗2)univ(µn)) = 0 so there exists λn−1∈Λ²(D_univ)such that

µ_n=−1

6∂²(λn−1) =−1

6|[r, λ_n−1]|

which gives the induction step and allows us to construct J.

Remark 3.7. Following Enriquez’s proof, it seems that the term Jn−1 in the~-series ofJ is built from termsΦn−1andΦnin the~-series ofΦ, as we had to correctJn−1byλn−1 which seems to be dependent fromΦ_n. On the other hand, it is clear, from the Etingof-Kazhdan’s formula that theirJn−1

do not depend fromΦ_n. This is not surprising: in Enriquez’s construction, the correcting term λn−1 only depends on an anti-symmetrization of Φn

which is unique (it is an easy check).

4. A G∞-morphism between chains and tensor fields

4.1. A differential d⁰_T on Λ^·T_poly^⊗· and G∞-morphism ψ : (Λ^·T_poly^⊗·, d⁰_T)→(Λ^·D_poly^⊗·, d_D)

The objective of this section is to prove the following proposition:

Proposition 4.1. There exist a differential (and coderivation) d⁰_T on Λ^·Tpoly⊗· and a morphism of differential coalgebras ψ: (Λ^·Tpoly⊗·, d⁰_T) → (Λ^·Dpoly⊗·, dD) such that the induced map ψ¹ : Tpoly → Dpoly is the Hochschild-Kostant-Rosenberg map ϕ¹ of Section 0.

Proof. Fori=T orD andn≥0, let us set V_i^[n]= ^M

p1+···+p_k=n

g^⊗p_i ¹Λ· · ·Λg^⊗p_i ^k

and V_i^[≤n]=^P_k≤nV_i^[k]. Letd^p_D¹^,...,p^k:Dpoly⊗p1Λ· · ·ΛDpoly⊗p_k →Dpoly be the components of the differential dD defining theG∞-structure of Dpoly

(see Definition 2.6) and denote d^[n]_D and d^[≤n]_D the sums d^[n]_D = ^X

p1+···+p_k=n

d^p_D¹^,...,p^k and d^[≤n]_D = ^X

p≤n

d^[p]_D.