Ju n 20 07 Koszul duality in deformation quantization , I

Let α be a polynomial Poisson bivector on a finite-dimensional vector space V over C. Then Kontsevich [K97] gives a formula for a quantization f ⋆ g of the algebra S(V )∗. We give a construction of an algebra with the PBW property defined from α by generators and relations. Namely, we define an algebra as the quotient of the free tensor algebra T (V ∗) by relations xi ⊗ xj − xj ⊗ xi = Rij(~) where Rij(~) ∈ T (V ∗)⊗ ~C[[~]], Rij = ~Sym(αij) +O(~ ), with one relation for each pair of i, j = 1... dimV . We prove that the constructed algebra obeys the PBW property, and this is a generalization of the Poincaré-Birkhoff-Witt theorem. In the case of a linear Poisson structure we get the PBW theorem itself, and for a quadratic Poisson structure we get an object closely related to a quantum R-matrix on V . At the same time we get a free resolution of the deformed algebra (for an arbitrary α). The construction of this PBW algebra is rather simple, as well as the proof of the PBW property. The major efforts should be undertaken to prove the conjecture that in this way we get an algebra isomorphic to the Kontsevich star-algebra. 1 The main construction 1.1 First of all, recall here the Stasheff’s definition of the Hochschild cohomological complex of an associative algebra A. Consider the shifted vector space W = A[−1], and the cofree coassociative coalgebra C(W ) (co)generated by W . As a graded vector space, C(W ) = T (A[1]), the free tensor space. The coproduct is: ∆(a1 ⊗ a2 ⊗ · · · ⊗ ak) = k−1 ∑ i=1 (a1 ⊗ · · · ⊗ ai) ⊗ (ai+1 ⊗ · · · ⊗ ak) (1) Consider the Lie algebra CoDer(C(A[1])) of all coderivations of this coalgebra. As the coalgebra is free, any coderivation D (if it is graded) is uniquely defined by a map ΨD : A ⊗k → A, and the degree of this coderivation is k − 1 (in conditions that A is not graded). The bracket [ΨD1 ,ΨD2 ] is again a coderivation. Define the Hochschild Lie algebra as Hoch q (A) = CoDer q (C(A[1])). To define the complex structure on it, consider