Quantum differentiation and chain maps of bimodule complexes

TL;DR

This paper explores the Hochschild cohomology of skew group algebras formed by a finite group acting on a vector space, introducing explicit chain maps and quantum differentiation techniques to unify and extend existing results.

Contribution

It provides an explicit chain map between resolutions that unifies previous cohomology results and incorporates quantum differentiation via Demazure-BBG operators.

Findings

Explicit chain map from bar to Koszul resolution

Unification of Hochschild cohomology results

Introduction of quantum differentiation with Demazure-BBG operators

Abstract

We consider a finite group acting on a vector space and the corresponding skew group algebra generated by the group and the symmetric algebra of the space. This skew group algebra illuminates the resulting orbifold and serves as a replacement for the ring of invariant polynomials, especially in the eyes of cohomology. One analyzes the Hochschild cohomology of the skew group algebra using isomorphisms which convert between resolutions. We present an explicit chain map from the bar resolution to the Koszul resolution of the symmetric algebra which induces various isomorphisms on Hochschild homology and cohomology, some of which have appeared in the literature before. This approach unifies previous results on homology and cohomology of both the symmetric algebra and skew group algebra. We determine induced combinatorial cochain maps which invoke quantum differentiation (expressed by Demazure-BBG operators).