PBW deformations of skew group algebras in positive characteristic

TL;DR

This paper explores new PBW deformations of skew group algebras in positive characteristic, revealing unique structures not present in characteristic zero and connecting them to Hochschild cohomology.

Contribution

It introduces a novel class of PBW deformations in positive characteristic and establishes a general framework linking these deformations to Hochschild cohomology.

Findings

New deformations in positive characteristic do not mirror characteristic zero cases.

In characteristic zero, Lusztig-type deformations are always isomorphic to Drinfeld-type.

A double complex approach connects deformations with Hochschild cohomology.

Abstract

We investigate deformations of a skew group algebra that arise from a finite group acting on a polynomial ring. When the characteristic of the underlying field divides the order of the group, a new type of deformation emerges that does not occur in characteristic zero. This analogue of Lusztig's graded affine Hecke algebra for positive characteristic can not be forged from the template of symplectic reflection and related algebras as originally crafted by Drinfeld. By contrast, we show that in characteristic zero, for arbitrary finite groups, a Lusztig-type deformation is always isomorphic to a Drinfeld-type deformation. We fit all these deformations into a general theory, connecting Poincar\'e-Birkhoff-Witt deformations and Hochschild cohomology when working over fields of arbitrary characteristic. We make this connection by way of a double complex adapted from Guccione, Guccione, and Valqui, formed from the Koszul resolution of a polynomial ring and the bar resolution of a group algebra.