An $L_\infty$ algebra structure on polyvector fields

TL;DR

This paper constructs an $L_$ algebra structure on polyvector fields that extends the classical Lie algebra structure, addressing obstructions in the infinite-dimensional case and establishing quasi-isomorphisms with Hochschild cochains.

Contribution

It introduces a novel $L_$ algebra structure on polyvector fields with higher Taylor components, generalizing the Schouten-Nijenhuis bracket and relating it to Hochschild cochains.

Findings

Constructed an $L_$ algebra with higher Taylor components.

Proved the $L_$ algebra is quasi-isomorphic to Hochschild cochains.

Established the $L_$ algebra is quasi-isomorphic to polyvector fields in finite dimensions.

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 $L_\infty$ structure on polyvector fields on $V$ having the even degree Taylor components, with the degree 2 component given by the Schouten-Nijenhuis bracket, but having as well higher non-vanishing Taylor components. We prove that this $L_\infty$ algebra is quasi-isomorphic to the corresponding Hochschild cochain complex. We prove that our $L_\infty$ algebra is $L_\infty$ quasi-isomorphic to the Lie algebra of polyvector fields on $V$ with the Schouten-Nijenhuis bracket, if $V$ is finite-dimensional.