Morita equivalence of formal Poisson structures

TL;DR

This paper extends Morita equivalence to formal Poisson structures, providing a complete classification of deformations near the zero structure using B-field transformations, and linking Poisson geometry with noncommutative algebra.

Contribution

It introduces Morita equivalence for formal Poisson structures and characterizes deformations near zero using B-field transformations, connecting to deformation quantization.

Findings

Complete description of Morita equivalence for formal Poisson structures near zero

Classification of deformations using B-field transformations

Links between Poisson geometry and noncommutative algebra via deformation quantization

Abstract

We extend the notion of Morita equivalence of Poisson manifolds to the setting of {\em formal} Poisson structures, i.e., formal power series of bivector fields $\pi=\pi_0 + \lambda\pi_1 +\cdots$ satisfying the Poisson integrability condition $[\pi,\pi]=0$. Our main result gives a complete description of Morita equivalent formal Poisson structures deforming the zero structure ($\pi_0=0$) in terms of $B$-field transformations, relying on a general study of formal deformations of Poisson morphisms and dual pairs. Combined with previous work on Morita equivalence of star products, our results link the notions of Morita equivalence in Poisson geometry and noncommutative algebra via deformation quantization.