An $$L_\infty $$L∞ algebra structure on polyvector fields

Abstract

It is well-known that the Kontsevich formality [K97] for Hochschild cochains of the polynomial algebra A = S(V * ) fails if the vector space V is infinite-dimensional. In the present paper, we study the corresponding obstructions. We construct an L ∞ structure on polyvector fields on V having the even degree Taylor components. The degree 2 component is given by the Schouten-Nijenhuis bracket, but all its higher even degree components are nonzero. We prove that this L ∞ algebra is quasi-isomorphic to the corresponding Hochschild cochain complex. We prove that our L ∞ algebra is L ∞ quasi-isomorphic to the Lie algebra of polyvector fields on V with the Schouten-Nijenhuis bracket, if V is finite-dimensional.

An Explicit Two Step Quantization of Poisson Structures and Lie Bialgebras

“…. , ] 2k : T poly (R n ) ⊗2k → T poly (R n )[3 − 4k] k≥1 which we call a Kontsevich-Shoikhet Lie ∞ structure as it was was introduced by Boris Shoikhet in [Sh2] with a reference to an important contribution by Maxim Kontsevich via an informal communication. As the Schouten bracket, this Lie ∞ structure makes sense in infinite dimensions.…”

“…was constructed in [Sh2] for any n (including the case n = +∞) with the help of the hyperbolic geometry and transcendental formulae. It was shown in [W3, B] that this universal map (which comes in fact from a Lie ∞ morphism) is essentially unique and can, in fact, be constructed by a trivial (in the sense that no choice of an associator is needed) induction.…”

“…Up to gauge equivalence, this new Maurer-Cartan element Υ is the only non-trivial deformation of the standard Maurer-Cartan element • • G G . We call this element Kontsevich-Shoikhet one as it was first found by Boris Shoikhet in [Sh2] with a reference to an important contribution by Maxim Kontsevich via an informal communication.…”

“…It was introduced by Boris Shoikhet in [Sh2] with a reference to an important contribution by Maxim Kontsevich via a private communication. This Lie ∞ structure is well defined in the limit n → +∞.…”

“…Example: a Kontsevich-Shoikhet Lie ∞ structure. If d = 2, then only oriented graphs Γ with even number 2p of vertices contribute into Υ g , and the leading terms are given explicitly by [Sh2] (25)…”