A Poincare-Birkhoff-Witt theorem for quadratic algebras with group actions

TL;DR

This paper extends the Poincare-Birkhoff-Witt theorem to quadratic algebras with group actions, providing new conditions for the PBW property in the context of skew group algebras and their deformations.

Contribution

It generalizes previous PBW criteria to nonsemisimple group algebras acting on Koszul algebras, using a double complex approach for analysis.

Findings

Established PBW conditions for skew group algebras in arbitrary characteristic.

Applied results to graded Hecke and Drinfeld orbifold algebras.

Provided a practical method for analyzing Hochschild cohomology and deformations.

Abstract

Braverman and Gaitsgory gave necessary and sufficient conditions for a nonhomogeneous quadratic algebra to satisfy the Poincare-Birkhoff-Witt property when its homogeneous version is Koszul. We widen their viewpoint and consider a quotient of an algebra that is free over some (not necessarily semisimple) subalgebra. We show that their theorem holds under a weaker hypothesis: We require the homogeneous version of the nonhomogeneous quadratic algebra to be the skew group algebra (semidirect product algebra) of a finite group acting on a Koszul algebra, obtaining conditions for the Poincare-Birkhoff-Witt property over (nonsemisimple) group algebras. We prove our main results by exploiting a double complex adapted from Guccione, Guccione, and Valqui (formed from a Koszul complex and a resolution of the group), giving a practical way to analyze Hochschild cohomology and deformations of skew group algebras in positive characteristic. We apply these conditions to graded Hecke algebras and Drinfeld orbifold algebras (including rational Cherednik algebras and symplectic reflection algebras) in arbitrary characteristic, with special interest in the case when the characteristic of the underlying field divides the order of the acting group.