PBW deformations of skew polynomial rings and their group extensions

TL;DR

This paper studies PBW deformations of skew polynomial rings extended by finite groups, providing homological criteria for their structure and linking certain cases to generalized enveloping algebras of color Lie algebras.

Contribution

It offers a homological interpretation of PBW deformation conditions and characterizes when these deformations correspond to generalized enveloping algebras.

Findings

Homological criteria for PBW deformations via Gerstenhaber brackets

Conditions for deformations to be generalized enveloping algebras

Clarification of parameter functions for quantum Drinfeld orbifold algebras

Abstract

We examine PBW deformations of finite group extensions of skew polynomial rings, in particular the quantum Drinfeld orbifold algebras defined by the first author. We give a homological interpretation, in terms of Gerstenhaber brackets, of the necessary and sufficient conditions on parameter functions to define a quantum Drinfeld orbifold algebra, thus clarifying the conditions. In case the acting group is trivial, we determine conditions under which such a PBW deformation is a generalized enveloping algebra of a color Lie algebra; our PBW deformations include these algebras as a special case.