Weak quantization of Poisson structures

TL;DR

This paper proves that Poisson structures on sheaves of Lie algebroids can be weakly quantized and provides conditions for full quantization, extending known results in complex symplectic geometry.

Contribution

It establishes the existence of weak deformation quantizations for Poisson structures on sheaves of Lie algebroids and offers criteria for their actual quantization, advancing deformation theory.

Findings

Any Poisson structure admits a weak deformation quantization.

Sufficient conditions for actual deformation quantization are provided.

Connections to known results in complex symplectic geometry are clarified.

Abstract

In this paper we prove that any Poisson structure on a sheaf of Lie algebroids admits a weak deformation quantization, and give a sufficient condition for such a Poisson structure to admit an actual deformation quantization. We also answer the corresponding classification problems. In the complex symplectic case, we recover in particular some results of Nest-Tsygan and Polesello-Schapira. We begin the paper with a recollection of known facts about deformation theory of cosimplicial differential graded Lie algebras.