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A Poincaré-Birkhoff-Witt theorem for quadratic algebras with group actions


Authors: Anne V. Shepler and Sarah Witherspoon
Journal: Trans. Amer. Math. Soc. 366 (2014), 6483-6506
MSC (2010): Primary 16S37, 16E40, 16S80, 16S35
DOI: https://doi.org/10.1090/S0002-9947-2014-06118-7
Published electronically: July 21, 2014
MathSciNet review: 3267016
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Abstract: Braverman, Gaitsgory, Polishchuk, and Positselski gave necessary and sufficient conditions for a nonhomogeneous quadratic algebra to satisfy the Poincaré-Birkhoff-Witt property when its homogeneous version is Koszul. We widen their viewpoint and consider a quotient of an algebra that is free over some (not necessarily semisimple) subalgebra. We show that their theorem holds under a weaker hypothesis: We require the homogeneous version of the nonhomogeneous quadratic algebra to be the skew group algebra (semidirect product algebra) of a finite group acting on a Koszul algebra, obtaining conditions for the Poincaré-Birkhoff-Witt property over (nonsemisimple) group algebras. We prove our main results by exploiting a double complex adapted from Guccione, Guccione, and Valqui (formed from a Koszul complex and a resolution of the group), giving a practical way to analyze Hochschild cohomology and deformations of skew group algebras in positive characteristic. We apply these conditions to graded Hecke algebras and Drinfeld orbifold algebras (including rational Cherednik algebras and symplectic reflection algebras) in arbitrary characteristic, with special interest in the case when the characteristic of the underlying field divides the order of the acting group.


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Additional Information

Anne V. Shepler
Affiliation: Department of Mathematics, University of North Texas, Denton, Texas 76203
Email: ashepler@unt.edu

Sarah Witherspoon
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
Email: sjw@math.tamu.edu

DOI: https://doi.org/10.1090/S0002-9947-2014-06118-7
Keywords: Koszul algebras, skew group algebras, Hochschild cohomology, Drinfeld orbifold algebras
Received by editor(s): September 25, 2012
Received by editor(s) in revised form: January 17, 2013
Published electronically: July 21, 2014
Additional Notes: The first author was partially supported by NSF grants #DMS-0800951 and #DMS-1101177
The second author was partially supported by NSF grants #DMS-0800832 and #DMS-1101399
Article copyright: © Copyright 2014 American Mathematical Society