Lift of C_∞ and L_∞ morphisms to G_∞ morphisms

arXiv:math/0304004v2 [math.QA] 29 Jun 2005 LIFTS OF C∞ AND L∞ -MORPHISMS TO G∞ -MORPHISMS Grégory Ginot(a) , Gilles Halbout(b) (a) Laboratoire Analyse Géométrie et Applications, Université Paris 13 et Ecole Normale Supèrieure de Cachan, France (b) Institut de Recherche Mathématique Avancée, Université Louis Pasteur et CNRS, Strasbourg, France Abstract. Let g2 be the Hochschild complex of cochains on C ∞ (Rn ) and g1 be the space of multivector fields on Rn . In this paper we prove that given any G∞ -structure (i.e. Gerstenhaber algebra up to homotopy structure) on g2 , and any C∞ -morphism ϕ (i.e. morphism of commutative, associative algebra up to homotopy) between g1 and g2 , there exists a G∞ morphism Φ between g1 and g2 that restricts to ϕ. We also show that any L∞ -morphism (i.e. morphism of Lie algebra up to homotopy), in particular the one constructed by Kontsevich, can be deformed into a G∞ -morphism, using Tamarkin’s method for any G∞ -structure on g2 . We also show that any two of such G∞ -morphisms are homotopic. 0-Introduction Let M be a differential manifold and g2 = (C • (A, A), b) be the Hochschild cochain complex on A = C ∞ (M ). The classical Hochschild-Kostant-Rosenberg theorem states that the cohomology of g2 is the graded Lie algebra g1 = Γ(M, ∧• T M ) of multivector fields on M . There is also a graded Lie algebra structure on g2 given by the Gerstenhaber bracket. In particular g1 and g2 are also Lie algebras up to homotopy (L∞ -algebra for short). In the case M = Rn , using different methods, Kontsevich ([Ko1] and [Ko2]) and Tamarkin ([Ta]) have proved the existence of Lie homomorphisms “up to homotopy” (L∞ -morphisms) from g1 to g2 . Kontsevich’s proof uses graph complex and is related to multizeta functions whereas Tamarkin’s construction uses the existence of Drinfeld’s associators. In fact Tamarkin’s L∞ -morphism comes from the restriction of a Gerstenhaber algebra up to homotopy homomorphism (G∞ -morphism) from g1 to g2 . The G∞ -algebra structure on g1 is induced by its classical Gerstenhaber algebra structure and a far less trivial G∞ -structure on g2 was proved to exist by Tamarkin [Ta] and relies on a Drinfeld’s associator. Tamarkin’s G∞ -morphism also restricts into a commutative, associative up to homotopy morphism (C∞ -morphism for short). The C∞ -structure on g2 (given by Keywords : Deformation quantization, star-product, homotopy formulas, homological methods. AMS Classification : Primary 16E40, 53D55, Secondary 18D50, 16S80. Typeset by AMS-TEX 1 2 GREGORY GINOT, GILLES HALBOUT restriction of the G∞ -one) highly depends on Drinfeld’s associator, and any two choices of a Drinfeld associator yields a priori different C∞ -structures. When M is a Poisson manifold, Kontsevich and Tamarkin homomorphisms imply the existence of a star-product (see [BFFLS1] and [BFFLS2] for a definition). A connection between the two approaches has been given in [KS] but the morphisms given by Kontsevich and Tamarkin are not the same. The aim of this paper is to show that, given any G∞ -structure on g2 and any C∞ -morphism ϕ between g1 and g2 , there exists a G∞ -morphism Φ between g1 and g2 that restricts to ϕ. We also show that any L∞ -morphism can be deformed into a G∞ -one. In the first section, we fix notation and recall the definitions of L∞ and G∞ -structures. In the second section we state and prove the main theorem. In the last section we show that any two G∞ -morphisms given by Tamarkin’s method are homotopic. Remark : In the sequel, unless otherwise is stated, the manifold M is supposed to be Rn for some n ≥ 1. Most results could be generalized to other manifolds using techniques of Kontsevich [Ko1] (also see [TS], [CFT]). 1-C∞ , L∞ and G∞ -structures For any graded vector space g, we choose the following degree on ∧• g : if X1 , . . . , Xk are homogeneous elements of respective degree |X1 |, . . . |Xk |, then |X1 ∧ · · · ∧ Xk | = |X1 | + · · · + |Xk | − k. In particular the component g = ∧1 g ⊂ ∧• g is the same as the space g with degree shifted by one. The space ∧• g with the deconcatenation cobracket is the cofree cocommutative coalgebra on g with degree shifted by one (see [LS], Section 2). Any degree one map dk : ∧k g → g (k ≥ 1) extends into a derivation dk : ∧• g → ∧• g of the coalgebra ∧• g by cofreeness property. Definition 1.1. A vector space g is endowed with a L∞ -algebra (Lie algebras “up to homotopy”) structure if there are degree one linear maps m1,...,1 , with k ones : ∧k g → g such that if we extend them to maps ∧• g → ∧• g, then d ◦ d = 0 where d is the derivation d = m1 + m1,1 + · · · + m1,...,1 + · · · . For more details on L∞ -structures, see [LS]. It follows from the definition that a L∞ -algebra structure induces a differential coalgebra structure on ∧• g and that the map m1 : g → g is a differential. If m1,...,1 : ∧k g → g are 0 for k ≥ 3, we get the usual definition of (differential if m1 6= 0) graded Lie algebras. For any graded vector space g, we denote g⊗n the quotient of g⊗n by the image of all shuffles of length n (see [GK] or [GH] for details). The graded vector space ⊕n≥0 g⊗n is a quotient coalgebra of the tensor coalgebra ⊕n≥0 g⊗n . It is well known that this coalgebra ⊕n≥0 g⊗n is the cofree Lie coalgebra on the vector space g (with degree shifted by minus one). Definition 1.2. A C∞ -algebra (commutative and asssociative “up to homotopy” algebra) structure on a vector space g is given by a collection of degree one linear maps mk : g⊗k → g such that if we extend them to maps ⊕g⊗• → ⊕g⊗• , then d ◦ d = 0 where d is the derivation d = m1 + m2 + m3 + · · · . In particular a C∞ -algebra is an A∞ -algebra. LIFT OF C∞ AND L∞ MORPHISMS TO G∞ MORPHISMS 3 For any space g, we denote ∧• g⊗• the graded space ∧• g⊗• = ⊕ m≥1, p1 +···+pn =m g⊗p1 ∧ · · · ∧ g⊗pn . We use the following grading on ∧• g⊗• : for x11 , · · · , xpnn ∈ g, we define |x11 ⊗···⊗ xp11 ∧···∧ x1n ⊗···⊗ xpnn | = p1 X |xi11 | +···+ i1 p1 X |xinn | − n. in Notice that the induced grading on ∧• g ⊂ ∧• g⊗• is the same than the one introduced above. The cobracket on ⊕g⊗• and the coproduct on ∧• g extend to a cobracket and a coproduct on ∧•g⊗• which yield a Gerstenhaber coalgebra structure on ∧• g⊗• . It is well known that this coalgebra structure is cofree (see [Gi], Section 3 for example). Definition 1.3. A G∞ -algebra (Gerstenhaber algebra “up to homotopy”) structure on a graded vector space g is given by a collection of degree one maps mp1 ,...,pn : g⊗p1 ∧ · · · ∧ g⊗pn → g indexed by p1 , . . . pn ≥ 1 such that their canonical extension: ∧• g⊗• → ∧• g⊗• satisfies d ◦ d = 0 where X d= mp1 ,...,pn . m≥1, p1 +···pn =m Again, as the coalgebra structure of ∧• g⊗• is cofree, the map d makes ∧• g⊗• into a differential coalgebra. If the maps mp1 ,...,pn are 0 for (p1 , p2 , . . . ) 6= (1, 0, . . . ), (1, 1, 0, . . . ) or (2, 0, . . . ), we get the usual definition of (differential if m1 6= 0) Gerstenhaber algebra. The space of multivector fields g1 is endowed with a graded Lie bracket [−, −]S called the Schouten bracket (see [Kos]). This Lie algebra can be extended into a Gerstenhaber algebra, with commutative structure given by the exterior product: (α, β) 7→ α ∧ β 1,1 ⊗2 2 2 2 Setting d1 = m1,1 1 + m1 , where m1 : ∧ g1 → g1 , and m1 : g1 → g1 are the extension of the Schouten bracket and the exterior product, we find that (g1 , d1 ) is a G∞ -algebra. In the sameL way, one can define a differential Lie algebra structure on the vector space g2 = C(A, A) = k≥0 C k (A, A), the space of Hochschild cochains (generated by differential klinear maps from Ak to A), where A = C ∞ (M ) is the algebra of smooth differential functions over M . Its bracket [−, −]G , called the Gerstenhaber bracket, is defined, for D, E ∈ g2 , by |E||D| [D, E]G = {D|E} − (−1) {E|D}, where {D|E}(x1 , . . ., xd+e−1 ) = X (−1) |E|·i D(x1 , . . ., xi , E(xi+1 , . . ., xi+e ), . . .). i≥0 The space g2 has a grading defined by | D |= k ⇔ D ∈ C k+1 (A, A) and its differential is b = [m, −]G , where m ∈ C 2 (A, A) is the commutative multiplication on A. 4 GREGORY GINOT, GILLES HALBOUT Tamarkin (see [Ta] or also [GH]) stated the existence of a G∞ -structure on g2 (depending 2 on a choice of a Drinfeld associator) given by a differential d2 = m12 + m1,1 2 + m2 + · · · + p1 ,...,pn ⊗• m2 + · · · , on ∧•g2 satisfying d2 ◦ d2 = 0. Although this structure is non-explicit, it satisfies the following three properties : (a) m12 is the extension of the differential b (b) m1,1 2 is the extension of the Gerstenhaber bracket [−, −]G =0 and m1,1,...,1 2 (c) m22 induces the exterior product in cohomology and the collection of the (mk )k≥1 defines a C∞ -structure on g2 .(1.1) Definition 1.4. A L∞ -morphism between two L∞ -algebras (g1 , d1 = m11 + . . . ) and (g2 , d2 = m12 + . . . ) is a morphism of differential coalgebras ϕ : (∧• g1 , d1 ) → (∧• g2 , d2 ). (1.2) Such a map ϕ is uniquely determined by a collection of maps ϕn : ∧n g1 → g2 (again by cofreeness properties). In the case g1 and g2 are respectively the graded Lie algebra (Γ(M, ∧T M ), [−, −]S ) and the differential graded Lie algebra (C (A, A) , [−, −]G ), the formality theorems of Kontsevich and Tamarkin state the existence of a L∞ -morphism between g1 and g2 such that ϕ1 is the Hochschild-Kostant-Rosenberg quasi-isomorphism. Definition 1.5. A morphism of C∞-algebras between two C∞-algebras (g1 , d1 ) and (g2 , d2 ) ⊗• is a map φ : (⊕g⊗• 1 , d1 ) → (⊕g2 , d2 ) of codifferential coalgebras. A C∞ -morphism is in particular a morphism of A∞ -algebras and is uniquely determined by maps ∂ k : g⊗k → g. Definition 1.6. A morphism of G∞-algebras between two G∞-algebras (g1 , d1 ) and (g2 , d2 ) • ⊗• is a map φ : (∧•g⊗• 1 , d1 ) → (∧ g2 , d2 ) of codifferential coalgebras. There are coalgebras inclusions ∧• g → ∧• g⊗• , ⊕g⊗• → ∧• g⊗• and it is easy to check P p1 ,...,pn P ′ p1 ,...,pn
′ that any G∞ -morphism between two G∞ -algebras (g, m ), g , m re P 1,...,1
P ′ 1,...,1  • • ′ stricts to a L∞ -morphism ∧ g, m → ∧ g, m and a C∞ -morphism  
P ⊗• P k ⊕g⊗• , mk → ⊕g′ , m′ . In the case g1 and g2 are as above, Tamarkin’s theorem states that there exists a G∞ -morphism between the two G∞ algebras g1 and g2 (with the G∞ structure he built) that restricts to a C∞ and a L∞ -morphism. 2-Main theorem We keep the notations of the previous section, in particular g2 is the Hochschild complex of cochains on C ∞ (M ) and g1 its cohomology. Here is our main theorem. LIFT OF C∞ AND L∞ MORPHISMS TO G∞ MORPHISMS 5 Theorem 2.1. Given any G∞ -structure d2 on g2 satisfying the three properties of (1.1), and any C∞ -morphism ϕ between g1 and g2 such that ϕ1 is the Hochschild-KostantRosenberg map, there exists a G∞ -morphism Φ : (g1 , d1 ) → (g2 , d2 ) that restricts to ϕ. Also, given any L∞ -morphism γ between g1 and g2 such that γ 1 is the HochschildKostant-Rosenberg map, there exists a G∞ -structure (g1 , d′1 ) on g1 and G∞ -morphism Γ : (g1 , d′1 ) → (g2 , d2 ) that restricts to γ. Moreover there exists a G∞ -morphism Γ′ : (g1 , d1 ) → (g1 , d1 ). In particular, Theorem 2.1 applies to the formality map of Kontsevich and also to any C∞ -map derived (see [Ta], [GH]) from any B∞ -structure on g2 lifting the Gerstenhaber structure of g1 . Let us first recall the proof of Tamarkin’s formality theorem (see [GH] for more details): 1. First one proves there exists a G∞ -structure on g2 , with differential d2 , as in (1.1). 2. Then, one constructs a G∞ -structure on g1 given by a differential d′1 together with a G∞ -morphism Φ between (g1 , d′1 ) and (g2 , d2 ). 3. Finally, one constructs a G∞ -morphism Φ′ between (g1 , d1 ) and (g1 , d′1 ). The composition Φ ◦ Φ′ is then a G∞ -morphism between (g1 , d1 ) and (g2 , d2 ),thus restricts to a L∞ -morphism between the differential graded Lie algebras g1 and g2 . We suppose now that, in the first step, we take any G∞ -structure on g2 given by a differential d2 and we suppose we are given a C∞ -morphism ϕ and a L∞ -morphism γ between g1 and g2 satisfying γ 1 = ϕ1 = ϕHKR the Hochschild-Kostant-Rosenberg quasi-isomorphism. Proof of Theorem 2.1: The Theorem will follow if we prove that steps 2 and 3 of Tamarkin’s construction are still true with the extra conditions that the restriction of the G∞ -morphism Φ (resp. Φ′ ) on the C∞ -structures is the C∞ -morphism ϕ : g1 → g2 (resp. id). Let us recall (see [GH]) that the constructions of Φ and d′1 can be made by induction. For i = 1, 2 and n ≥ 0, let us set M [n] Vi = gi⊗p1 ∧ · · · ∧ gi⊗pk p1 +···+pk =n [≤n] and Vi = P [k] [n] p1 +···+pk =n Clearly, d2 = [≤n] Vi . Let d2 and d2 be the sums X X [p] [n] [≤n] d2 = dp21 ,...,pk and d2 = d2 . k≤n P n≥1 [n] d′ 1 p≤n P [n] [n] d2 . In the same way, we denote d′1 = n≥1 d′ 1 with X X ′ [k] [≤n] = d1p1 ,...,pk and d′ 1 = d′ 1 . p1 +···+pk =n 1≤k≤n ′ • ⊗• We know from Section 1 that a morphism Φ : (∧•g⊗• 1 , d1 ) → (∧ g2 , d2 ) is uniquely determined by its components Φp1 ,...,pk : g1⊗p1 ∧ · · · ∧ g1⊗pk → g2 . Again, we have Φ = P [n] with n≥1 Φ X X Φ[n] = Φp1 ,...,pk and Φ[≤n] = Φ[k] . p1 +···+pk =n 1≤k≤n 6 GREGORY GINOT, GILLES HALBOUT [n] We want to construct the maps d′ 1 and Φ[n] by induction with the initial condition [1] d′ 1 = 0 Φ[1] = ϕHKR , and where ϕHKR : (g1 , 0) → (g2 , b) is the Hochschild-Kostant-Rosenberg quasi-isomorphism (see [HKR]) defined, for α ∈ g1 , f1 , · · · , fn ∈ A, by
ϕHKR : α 7→ (f1 , . . . , fn ) 7→ hα, df1 ∧ · · · ∧ dfn i . Moreover, we want the following extra conditions to be true: 2 k≥3 d′ 1 = d21 , Φk≥2 = ϕk , d′ 1 = 0. (2.3) [i] Now suppose the construction is done for n−1 (n ≥ 2), i.e., we have built maps (d′ 1 )i≤n−1 and (Φ[i] )i≤n−1 satisfying conditions (2.3) and [≤n−1] Φ[≤n−1] ◦ d′ 1 [≤n−1] = d2 [≤n−1] [≤n−1] and d′ 1 and Φ′ ◦ Φ[≤n−1] on V1 [≤n−1] ◦ d′ 1 [≤n] = 0 on V1 . (2.4) [i] [n] In [GH], we prove that for any such (d′ 1 )i≤n−1 and (Φ[i] )i≤n−1 , one can construct d′ 1 and Φ[n] such that condition (2.4) is true for n instead of n − 1. To complete the proof [n] of Theorem 2.1 (step 2), we have to show that d′ 1 and Φ[n] can be chosen to satisfy k conditions (2.3). In the equation 2.4, the terms d′ 1 and Φk only act on V1k . So one can 2 i replace Φn with ϕn , d′ 1 with d21 (or d′ 1 , i ≥ 3 with 0) provided conditions (2.4) are still satisfied on V1n . The other terms acting on V1n in the equation (2.4) only involve terms m Φm = ϕm and d′ 1 . Then conditions (2.4) on V11,...,1 are the equations that should be P 1,1 k satisfied by a C∞ -morphism between the C∞ -algebras (g1 , d′ 1 = d1,1 1 ) and (g2 , k≥1 d2 ) restricted to V1n . Hence by hypothesis on ϕ the conditions hold. Similarly the construction of Φ′ can be made by induction. Let us recall the proof given • ⊗• ′ in [GH]. Again a morphism Φ′ : (∧•g⊗• 1 , d1 ) → (∧ g2 , d1 ) is uniquely determined by its P [n] p ,...,pk components Φ′ 1 : g1⊗p1 ∧ · · · ∧ g1⊗pk → g1 . We write Φ′ = n≥1 Φ′ with Φ′ [n] = X Φ′ p1 ,...,pk [≤n] X = p1 +···+pk =n [k] Φ′ . 1≤k≤n [n] We construct the maps Φ′ by induction with the initial condition Φ′ we want the following extra conditions to be true: [1] = id. Moreover, n Φ′ = 0 for n ≥ 2. (2.5) [i] Now suppose the construction is done for n−1 (n ≥ 2), i.e., we have built maps (Φ′ )i≤n−1 satisfying conditions (2.5) and Φ′ [≤n−1] [≤n] d1 = d′1 [≤n] Φ′ [≤n−1] [≤n] on V1 . (2.6) LIFT OF C∞ AND L∞ MORPHISMS TO G∞ MORPHISMS [i] 7 [n] In [GH], we prove that for any such (Φ′ )i≤n−1 , one can construct Φ′ such that condition (2.6) is true for n instead of n − 1 in the following way : making the equation Φ′ d1 = d′1 Φ′ [n+1] on V1 explicit, we get Φ′ [≤n] [≤n+1] d1 = d′1 [≤n+1] [i] Φ′ [≤n] (2.7) . [n+1] [1] If we now take into account that d1 = 0 for i 6= 2, d′1 = 0 and that on V1 [k] [l] [≤k] ′ [l] Φ′ d1 = d′1 Φ = 0 for k + l > n + 2, the identity (2.7) becomes Φ ′ [≤n] [2] d1 = n+1 X d′1 [k] Φ′ [≤n−k+2] we have . k=2 As d′1 [2] = d1 [2] , (2.7) is equivalent to [2] ′ [≤n] d1 Ψ −Φ ′ [≤n] d1 [2] h [2] = d1 , Φ ′ [≤n] i =− n+1 X [k] d′1 Φ′ [≤n−k+2] . k=3 [2] 2 Notice that d1 = m1,1 1 + m1 . Then the construction will be possible when the term Pn+1 ′ [k] ′ [≤n−k+2] [2] is a couboundary in the subcomplex of (End(∧•g⊗• 1 ), [d1 , −]) conk=3 d1 ψ sisting of maps which restrict to zero on ⊕n≥2 g1 ⊗n . It is always a cocycle by straightforward computation (see [GH]) and the subcomplex is acyclic because both (End(∧•g⊗• 1 ), [2] [d1 , −]) and the Harrison cohomology of g1 are trivial according to Tamarkin [Ta] (see also [GH] Proposition 5.1 and [Hi] 5.4). In the case of the L∞ -morphism γ, the first step is similar: the fact that γ is a L∞ -map enables us to build a G∞ -structure (g1 , d′1 ) on g1 and a G∞ -morphism Γ : (g1 , d′1 ) → (g2 , d2 ) such that: Γ1,...,1 = γ 1,...,1 , 1,1 d′ 1 = d1,1 1 , 1,1,...,1 d′ 1 = 0. (2.8) For the second step, we have to build a map Γ′ satisfying the equation [2] ′ [≤n] d1 Γ ′ [≤n] −Γ d1 [2] n+1 h i X [k] [≤n−k+2] [2] ′ [≤n] = d1 , Γ =− d′1 Γ′ k=3 [n+1] on V1 for any n ≥ 1. Again, because Tamarkin has prooved that the complex [2] • ⊗• (End(∧ g1 ), [d1 , −]) is acyclic (we are in the case M = Rn ), the result follows from Pn+1 [k] [≤n−k+2] the fact that k=3 d′1 Γ′ is a cocycle. The difference with the C∞ -case is that ′ 1,...,1 the Γ could be non zero.  3-The difference between two G∞ -maps In this section we investigate the difference between two differents G∞ -formality maps. We fix once for all a G∞ -structure on g2 (given by a differential d2 ) satisfying the conditions (1.1) and a morphism of G∞ -algebras T : (g1 , d1 ) → (g2 , d2 ) such that T 1 : g1 → g2 is ϕHKR . Let K : (g1 , d1 ) → (g2 , d2 ) be any other G∞ -morphism with K 1 = ϕHKR (for example any lift of a Kontsevich formality map or any G∞ -maps lifting another C∞ morphism). 8 GREGORY GINOT, GILLES HALBOUT • ⊗• Theorem 3.1. There exists a map h : ∧•g⊗• 1 → ∧ g2 such that T − K = h ◦ d1 + d2 ◦ h. In other words the formality G∞ -morphisms K and T are homotopic.   • ⊗• , ∧ g ), δ with The maps T and K are elements of the cochain complex Hom(∧•g⊗• 2 1 • ⊗• differential given, for all f ∈ Hom(∧•g⊗• 1 , ∧ g2 ), |f | = k, by δ(f ) = d2 ◦ f − (−1)k f ◦ d1 .   ), [d ; −] and We first compare this cochain complex with the complexes End(∧•g⊗• 1 1   End(∧•g⊗• 2 ), [d2 ; −] (where [−; −] is the graded commutator of morphisms). There are morphisms • ⊗• • ⊗• ∗ • ⊗• • ⊗• • ⊗• T∗ : End(∧•g⊗• 1 ) → Hom(∧ g1 , ∧ g2 ), T : End(∧ g2 ) → Hom(∧ g1 , ∧ g2 ) • ⊗• • ⊗• defined, for f ∈ End(∧•g⊗• 2 ) and g ∈ Hom(∧ g1 , ∧ g2 ), by T∗ (f ) = T ◦ f, T ∗ (g) = g ◦ T. Lemma 3.2. The morphisms T∗ :  End(∧•g⊗• 1 ), [d1 ; −]      • ⊗• • ⊗• • ⊗• → Hom(∧ g1 , ∧ g2 ), δ ← End(∧ g2 ), [d2 ; −] : T ∗ of cochain complexes are quasi-isomorphisms. Remark: This lemma holds for every manifold M and any G∞ -morphism T : (g1 , d1 ) → (g2 , d2 ). Proof :. First we show that T∗ is a morphism of complexes. Let f ∈ End(∧•g⊗• 2 ) with |f | = k, then T∗ ([d1 ; f ]) =T ◦ d1 ◦ f − (−1)k T ◦ f ◦ d1 =d2 ◦ (T ◦ f ) − (−1)k (T ◦ f ) ◦ d1 =δ(T∗ (f )). Let us prove now that T∗ is a quasi-isomorphism. For any graded vector space g, the space ∧•g⊗• has the structure of a filtered space where the m-level of the filtration is F m (∧•g⊗• ) = ⊕p1 +···+pn −1≤m g⊗p1 ∧ . . . g⊗pn . Clearly the differential d1 and d2 are com• ⊗• • ⊗• • ⊗• • ⊗• patible with the filtrations on ∧•g⊗• 1 and ∧ g2 , hence End(∧ g1 ) and Hom(∧ g1 , ∧ g2 ) are filtered cochain complex. This yields two spectral sequences (lying in the first quade••,• which converge respectively toward the cohomology H • (End(∧•g⊗• )) rant) E••,• and E 1 • ⊗• and H • (Hom(∧•g⊗• , ∧ g )). By standard spectral sequence techniques it is enough to 1 2 LIFT OF C∞ AND L∞ MORPHISMS TO G∞ MORPHISMS prove that the map T∗0 quasi-isomorphism. 9 e •,• induced by T∗ on the associated graded is a : E0•,• → E 0 e •,• are respectively [d1 , −] = 0 and d1 ◦ (−) − (−) ◦ The induced differentials on E0•,• and E 1 2 0 d11 = b ◦ (−) where b is the Hochschild coboundary. By cofreeness property we have the following two isomorphisms E0•,• ∼ = End(g1 ), e •,• ∼ E 0 = Hom(g1 , g2 ). e 0 induced by T∗ is ϕ The map T∗0 : E0•,• → E •• HKR ◦ (−). Let p : g2 → g1 be the projection onto the cohomology, i.e. p ◦ ϕHKR = id. Let u : g1 → g2 be any map satisfying b(u) = 0 and set v = p ◦ u ∈ End(g1 ). One can choose a map w : g1 → g2 which satisfies for any x ∈ g1 the following identity ϕHKR ◦ p ◦ u(x) − u(x) = b ◦ w(x). It follows that ϕHKR (v) has the same class of homology as u which proves the surjectivity of T∗0 in cohomology. The identity p ◦ ϕHKR = id implies easily that T∗0 is also injective in cohomology which finish the proof of the lemma for T∗ . The proof that T ∗ is also a quasi-isomorphism is analogous. Proof of Theorem 3.1:.   • ⊗• • ⊗• It is easy to check that T − K is a cocycle in Hom(∧ g1 , ∧ g2 ), δ . The complex of     • ⊗• ∼ ), [d , −] , g ), [d , −] is trigraded with | |1 being the Hom(∧ g cochain End(∧•g⊗• = 1 1 1 1 1 ⊗p degree coming from the graduation of g1 and any element x lying in g1⊗p1 ∧ · · · ∧ g1 q satisfies |x|2 = q − 1,  |x|3 = p1 + . . . pq − q. In the case M = Rn , the cohomology  H • End(∧•g⊗• 1 ), [d1 , −] is concentrated in bidegree (| |2 , | |3 ) = (0, 0) (see [Ta], [Hi]). By   • ⊗• • ⊗• Lemma 3.2, this is also the case for the cochain complex Hom(∧ g1 , ∧ g2 ), δ . Thus, its cohomology classes are determined by complex morphisms (g1 , 0) → (g2 , d12 ) and it is enough to prove that T and K determine the same complex morphism (g1 , 0) → (g2 , d12 = b) which is clear because T 1 and K 1 are both equal to the Hochschild-Kostant-Rosenberg map.  Remark. It is possible to have an explicit formula for the map h in Theorem .3.1. In fact the quasi-isomorphism coming from Lemma 3.2 can be made explicit using explicit homotopy formulae for the Hochschild-Kostant-Rosenberg map (see [Ha] for example) and deformation retract techniques (instead of spectral sequences) as in [Ka]. The same techniques  explicit formulae for the quasi-isomorphism giving the acyclicity  also apply to give • ⊗• of End(∧ g1 ), [d1 ; −] in the proof of theorem 3.1 (see [GH] for example) 10 GREGORY GINOT, GILLES HALBOUT References [BFFLS1] F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, D. Sternheimer, Quantum mechanics as a deformation of classical mechanics, Lett. Math. 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Tamarkin, B Tsygan, Noncommutative differential calculus, homotopy BV algebras and formality conjectures, Methods Funct. Anal. Topology 6 (2000), 85–97. (a) Laboratoire Analyse G éométrie et Applications, Université Paris 13, Centre des Mathématiques et de Leurs Applications, ENS Cachan, 61, av. du Président Wilson, 94230 Cachan, France e-mail:[email protected] (b) Institut de Recherche Mathématique Avancée, Université Louis Pasteur et CNRS 7, rue René Descartes, 67084 Strasbourg, France e-mail:[email protected]