Symplectic algebroids, groupoid Toeplitz operators and deformation quantization

TL;DR

This paper develops a star-product on Poisson manifolds using Toeplitz operators associated with symplectic Lie algebroids and groupoids, extending previous symplectic case methods.

Contribution

It introduces a new approach to deformation quantization on Poisson manifolds via Toeplitz operators on groupoids with Heisenberg structures.

Findings

Defines a star-product on Poisson manifolds from symplectic Lie algebroids.

Generalizes Guillemin and Melrose's symplectic approach to a broader setting.

Connects Toeplitz operators, groupoids, and deformation quantization.

Abstract

We use Toeplitz operators to define a star-product on Poisson manifolds whose Poisson structure is induced by a symplectic Lie algebroid. The Toeplitz operators we consider are defined on groupoids whose algebroid can be endowed with a Heisenberg group structure on the fibers. This generalizes an approach due to Guillemin and Melrose in the symplectic case.