Deformations of linear Poisson orbifolds

TL;DR

This paper investigates deformations of linear Poisson orbifolds through quotient algebras of tensor algebras with group actions, establishing PBW properties and generalizing symplectic reflection algebras.

Contribution

It introduces a class of quotient algebras with PBW property that generalize symplectic reflection algebras for linear Poisson orbifolds.

Findings

Proves PBW property for quotient algebras of tensor algebras with group actions.

Defines generalized quadratic relations on tensor algebras.

Establishes these algebras as quantizations of Poisson structures.

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.