On Homological Reduction of Poisson Structures

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.