Vol. 5, No. 3, 2011

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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

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).

Hochschild cohomology, skew group algebra, Koszul resolution, Demazure–BGG operator, quantum differentiation
Mathematical Subject Classification 2000
Primary: 16E40
Secondary: 16S35
Received: 17 March 2010
Accepted: 11 June 2010
Published: 10 September 2011
Anne V. Shepler
Mathematics Department
University of North Texas
1155 Union Circle
Denton, TX 76203-1430
United States
Sarah Witherspoon
Department of Mathematics
Mailstop 3368
Texas A&M University
College Station, TX 77843-3368
United States