# Gerstenhaber brackets for skew group algebras in positive characteristic

## TL;DR

This paper develops new techniques using twisted product resolutions to evaluate Gerstenhaber brackets in skew group algebras over fields of positive characteristic, aiding deformation theory analysis.

## Contribution

It introduces a method for computing Gerstenhaber brackets via twisted product resolutions, simplifying the study of deformation theory in positive characteristic settings.

## Key findings

- Effective evaluation of Gerstenhaber brackets using twisted resolutions.
- Application to graded Hecke algebras in positive characteristic.
- Enhanced understanding of deformation theory for skew group algebras.

## Abstract

The deformation theory of an algebra is controlled by the Gerstenhaber bracket, a Lie bracket on Hochschild cohomology. We develop techniques for evaluating Gerstenhaber brackets of semidirect product algebras recording actions of finite groups over fields of positive characteristic. The Hochschild cohomology and Gerstenhaber bracket of these skew group algebras can be complicated when the characteristic of the underlying field divides the group order. We show how to investigate Gerstenhaber brackets using twisted product resolutions, which are often smaller and more convenient than the cumbersome bar resolution typically used. These resolutions provide a concrete description of the Gerstenhaber bracket suitable for exploring questions in deformation theory. We demonstrate with the prototypical example of a graded Hecke algebra (rational Cherednik algebra) in positive characteristic.