Vol. 5, No. 3, 2011

Download this article
Download this article For screen
For printing
Recent Issues

Volume 10
Issue 9, 1845–2052
Issue 8, 1601–1843
Issue 7, 1373–1600
Issue 6, 1147–1371
Issue 5, 939–1146
Issue 4, 695–938
Issue 3, 451–694
Issue 2, 215–450
Issue 1, 1–214

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 issues

The Journal
Cover
Editorial Board
Editors' Addresses
Editors' Interests
About the Journal
Scientific Advantages
Submission Guidelines
Submission Form
Subscriptions
Editorial Login
Contacts
Author Index
To Appear
 
ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Quantum differentiation and chain maps of bimodule complexes

Anne V. Shepler and Sarah Witherspoon

Vol. 5 (2011), No. 3, 339–360
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/