Poisson geometry and Morita equivalence

TL;DR

This paper explores the representation theory of Poisson manifolds, focusing on Morita equivalence and Picard groups, and discusses their connections with symplectic groupoids and Hamiltonian actions.

Contribution

It provides a comprehensive overview of geometric Morita theory for Poisson manifolds, integrating concepts from Dirac structures and twisted Poisson structures.

Findings

Link between Morita equivalence and symplectic groupoids

Introduction of Picard groups in Poisson geometry

Connections with Hamiltonian actions

Abstract

These notes discuss various aspect of the ``representation theory'' of Poisson manifolds, with focus on Morita equivalence and Picard groups. We give a brief introduction to Poisson geometry (including Dirac and twisted Poisson structures) and algebraic Morita theory before presenting the geometric Morita theory of Poisson manifolds. We also point out the connections with the theory of symplectic groupoids and hamiltonian actions.