On homological reduction of Poisson structures

TL;DR

This paper develops a homotopy Poisson algebra framework for Poisson reduction under group actions, generalizing classical models and providing new homological tools for reduced spaces in quasi-Poisson contexts.

Contribution

It introduces a homotopy Poisson algebra for Poisson reduction, extending classical BFV algebra, and constructs homological models for reduced quasi-Poisson spaces.

Findings

Established a homotopy Poisson algebra for equivariant Poisson reduction.

Derived homological models for reduced quasi-Poisson spaces.

Generalized classical BFV algebra to a homotopy setting.

Abstract

Given a $\mathfrak{g}$-action on a Poisson manifold $(M, \pi)$ and an equivariant map $J: M \rightarrow \mathfrak{h}^*,$ for $\mathfrak{h}$ a $\mathfrak{g}$-module, we obtain, under natural compatibility and regularity conditions previously considered by Cattaneo-Zambon, a homotopy Poisson algebra generalizing the classical BFV algebra described by Kostant-Sternberg in the usual hamiltonian setting. As an application of our methods, we also derive homological models for the reduced spaces associated to quasi-Poisson and hamiltonian quasi-Poisson spaces.