Deformation quantization and the action of Poisson vector fields

TL;DR

This paper investigates the conditions under which Lie algebra actions on Poisson manifolds can be extended to their deformation quantizations, introducing homological obstructions that determine the possibility of such extensions.

Contribution

It develops a homological framework to identify obstructions to lifting Lie algebra actions to deformed algebras in the context of deformation quantization.

Findings

Homological obstructions are identified for extending Lie algebra actions.

Obstructions vanish if and only if the action can be extended to the deformed algebra.

The approach generalizes previous results for one-dimensional Lie algebras.

Abstract

As one knows, for every Poisson manifold $M$ there exists a formal noncommutative deformation of the algebra of functions on it; it is determined in a unique way (up to an equivalence relation) by the given Poisson bivector. Let a Lie algebra $\mathfrak g$ act by derivations on the functions on $M$. The main question, which we shall address in this paper is whether it is possible to lift this action to the derivations on the deformed algebra. It is easy to see, that when dimension of $\mathfrak g$ is $1$, the only necessary and sufficient condition for this is that the given action is by Poisson vector fields. However, when dimension of $\mathfrak g$ is greater than $1$, the previous methods do not work. In this paper we show how one can obtain a series of homological obstructions for this problem, which vanish if there exists the necessary extension.