Drinfeld Orbifold Algebras

TL;DR

This paper introduces Drinfeld orbifold algebras, a broad class of filtered algebras deforming skew group algebras, unifying several important algebraic structures and providing criteria for their construction and classification.

Contribution

It defines Drinfeld orbifold algebras, establishes conditions for their existence, and connects their properties to Hochschild cohomology and PBW criteria, with applications to abelian groups.

Findings

Necessary and sufficient conditions for algebra construction

Explicit connection between Hochschild cohomology and PBW property

Classification of deformations arising as Drinfeld orbifold algebras

Abstract

We define Drinfeld orbifold algebras as filtered algebras deforming the skew group algebra (semi-direct product) arising from the action of a finite group on a polynomial ring. They simultaneously generalize Weyl algebras, graded (or Drinfeld) Hecke algebras, rational Cherednik algebras, symplectic reflection algebras, and universal enveloping algebras of Lie algebras with group actions. We give necessary and sufficient conditions on defining parameters to obtain Drinfeld orbifold algebras in two general formats, both algebraic and homological. We explain the connection between Hochschild cohomology and a Poincare-Birkhoff-Witt property explicitly (using Gerstenhaber brackets). We also classify those deformations of skew group algebras which arise as Drinfeld orbifold algebras and give applications for abelian groups.