An L∞ algebra structure on polyvector fields

In this paper we construct an L∞ structure on polyvector fields on a vector space V over C where V may be infinite-dimensional. We prove that the constructed L∞ algebra of polyvector fields is L∞ equivalent to the Hochschild complex of polynomial functions on V , even in the infinite-dimensional case. For a finite-dimensional space V , our L∞ algebra is equivalent to the classical Schouten-Nijenhuis Lie algebra of polyvector fields. For an infinite-dimensional V , it is essentially different. In particular, we get the higher obstructions for deformation quantization in infinitedimensional case. 1 The set-up Let V = ⊕i≥0Vi be an infinite-dimensional non-negatively graded vector space over C with finite-dimensional components Vi. Our goal in this paper is to establish the Kontsevich formality theorem for the Hochschild complex of the algebra S(V ∗) = S(⊕i≥0V ∗ i ). We start the paper with defining what are the right versions of polyvector fields on V and of the cohomological Hochschild complex of S(V ∗) for an infinite-dimensional vector space V with finite-dimensional graded components. Then we prove an analog of the Hochschild-Kostant-Rosenberg theorem in our setting. 1.1 The polyvector fields Tfin(V ) First of all, let us define an appropriate Lie algebra of polyvector fields Tpoly(V ). We want to allow some infinite sums. Here is the precise definition. An i-linear polyvector field γ of inner degree k is a (possibly infinite) sum in S(V ∗)⊗̂Λi(V ) such that all summands have degree k (here we set deg V ∗ a = −a). With a fixed element in Λ (V ), its ”coefficient” maybe only a finite sum because the vector space V is non-negatively graded. We denote the space of i-linear polyvector fields of inner degree k on the space V by T i,k fin (V ). Now the space of i-linear polyvector fields T i fin(V ) is by definition a direct sum T i fin(V ) = ⊕kT i,k fin (V ). That is, any polyvector field has a finite number of non-zero inner degrees. The Schouten-Nijenhuis bracket of two such polyvector fields is well-defined. We denote this graded Lie algebra by Tfin(V ).