On homological reduction of Poisson structures

Given a $\mathfrak{g}$-action on a Poisson manifold $(M,\pi )$ and an equivariant map $J:M\to {\mathfrak{h}}^{\ast },$ 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.