Group actions on algebras and the graded Lie structure of Hochschild cohomology

TL;DR

This paper investigates the graded Lie algebra structure of Hochschild cohomology for skew group algebras, focusing on deformation theory, bracket formulas, and applications to orbifold and representation theory.

Contribution

It provides new formulas and conditions for the Gerstenhaber bracket in Hochschild cohomology of skew group algebras, especially for abelian groups and polynomial actions.

Findings

Identified conditions when brackets vanish for polynomial skew group algebras.

Derived formulas expressing brackets via group character inner products.

Connected results to graded Hecke algebras and noncommutative Poisson structures.

Abstract

Hochschild cohomology governs deformations of algebras, and its graded Lie structure plays a vital role. We study this structure for the Hochschild cohomology of the skew group algebra formed by a finite group acting on an algebra by automorphisms. We examine the Gerstenhaber bracket with a view toward deformations and developing bracket formulas. We then focus on the linear group actions and polynomial algebras that arise in orbifold theory and representation theory; deformations in this context include graded Hecke algebras and symplectic reflection algebras. We give some general results describing when brackets are zero for polynomial skew group algebras, which allow us in particular to find noncommutative Poisson structures. For abelian groups, we express the bracket using inner products of group characters. Lastly, we interpret results for graded Hecke algebras.