Color Lie rings and PBW deformations of skew group algebras

TL;DR

This paper explores the relationship between color Lie rings over finite group algebras and their universal enveloping algebras, showing they can be viewed as PBW deformations of skew group algebras and are braided Hopf algebras.

Contribution

It establishes a correspondence between color Lie rings and quantum Drinfeld orbifold algebras, revealing their structure as PBW deformations and braided Hopf algebras.

Findings

Universal enveloping algebras are PBW deformations of skew group algebras.

Quantum Drinfeld orbifold algebras are braided Hopf algebras.

Every such algebra arises from a color Lie ring with a Yetter-Drinfeld structure.

Abstract

We investigate color Lie rings over finite group algebras and their universal enveloping algebras. We exhibit these universal enveloping algebras as PBW deformations of skew group algebras: Every color Lie ring over a finite group algebra with a particular Yetter-Drinfeld structure has universal enveloping algebra that is a quantum Drinfeld orbifold algebra. Conversely, every quantum Drinfeld orbifold algebra of a particular type arising from the action of an abelian group is the universal enveloping algebra of some color Lie ring over the group algebra. One consequence is that these quantum Drinfeld orbifold algebras are braided Hopf algebras.