Quantum differentiation and chain maps of bimodule complexes

Shepler, Anne; Witherspoon, Sarah

doi:10.2140/ant.2011.5.339

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Quantum differentiation and chain maps of bimodule complexes

Anne V. Shepler and Sarah Witherspoon

Vol. 5 (2011), No. 3, 339–360

DOI: 10.2140/ant.2011.5.339

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–BGG operators).

Keywords

Hochschild cohomology, skew group algebra, Koszul resolution, Demazure–BGG operator, quantum differentiation

Mathematical Subject Classification 2000

Primary: 16E40

Secondary: 16S35

Milestones

Received: 17 March 2010

Accepted: 11 June 2010

Published: 10 September 2011

Authors

Anne V. Shepler
Mathematics Department University of North Texas 1155 Union Circle Denton, TX 76203-1430 United States
http://www.math.unt.edu/~ashepler/
Sarah Witherspoon
Department of Mathematics Mailstop 3368 Texas A&M University College Station, TX 77843-3368 United States
http://www.math.tamu.edu/~sarah.witherspoon/

Vol. 5, No. 3, 2011