PBW Deformations of Skew Group Algebras in Positive Characteristic

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• 作者：Anne V. Shepler (1)
  Sarah Witherspoon (2)
  
  1. Department of Mathematics ; University of North Texas ; Denton ; TX ; 76203 ; USA
  2. Department of Mathematics ; Texas A & M University ; College Station ; TX ; 77843 ; USA
  
• 关键词：Graded affine Hecke algebra ; Drinfeld Hecke algebra ; Deformations ; Modular representations ; Hochschild cohomology ; 20C08 ; 20C20 ; 16E40
• 刊名：Algebras and Representation Theory
• 出版年：2015
• 出版时间：February 2015
• 年：2015
• 卷：18
• 期：1
• 页码：257-280
• 全文大小：420 KB
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• 刊物主题：Commutative Rings and Algebras; Associative Rings and Algebras; Non-associative Rings and Algebras;
• 出版者：Springer Netherlands
• ISSN：1572-9079

文摘

We investigate deformations of a skew group algebra that arise from a finite group acting on a polynomial ring. When the characteristic of the underlying field divides the order of the group, a new type of deformation emerges that does not occur in characteristic zero. This analogue of Lusztig鈥檚 graded affine Hecke algebra for positive characteristic can not be forged from the template of symplectic reflection and related algebras as originally crafted by Drinfeld. By contrast, we show that in characteristic zero, for arbitrary finite groups, a Lusztig-type deformation is always isomorphic to a Drinfeld-type deformation. We fit all these deformations into a general theory, connecting Poincar茅-Birkhoff-Witt deformations and Hochschild cohomology when working over fields of arbitrary characteristic. We make this connection by way of a double complex adapted from Guccione, Guccione, and Valqui, formed from the Koszul resolution of a polynomial ring and the bar resolution of a group algebra.