Deformations of linear Poisson orbifolds

Abstract

Let $\Gamma$ be a finite group acting faithfully and linearly on a vector space $V$. Let $T(V)$ ($S(V)$) be the tensor (symmetric) algebra associated to $V$ which has a natural $\Gamma$ action. We study generalized quadratic relations on the tensor algebra $T(V)\rtimes \Gamma$. We prove that the quotient algebras of $T(V)\rtimes \Gamma$ by such relations satisfy PBW property. Such quotient algebras can be viewed as quantizations of linear or constant Poisson structures on $S(V)\rtimes \Gamma$, and are natural generalizations of symplectic reflection algebras.