PBW Deformations of Skew Group Algebras in Positive Characteristic

Abstract

Abstract. 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's 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 Drinfeldtype deformation. We fit all these deformations into a general theory, connecting Poincaré-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.

PBW deformations of Koszul algebras over a nonsemisimple ring

“…Theorem 4.6.1, for the notions see Sect. 4.6), generalizing the corresponding results in [7,20,21,24].…”

“…We remind the reader that the PBW deformations appeared in [2,18,20,21,24] are more general in terms of generating relations. However, the situation in the theorem above occurs very common, and it applies to many interesting algebras, such as symplectic reflection algebras (cf.…”

“…[6,16,19]) for arbitrary finite group rings, and Lusztig-type algebras in positive characteristic (cf. [10], also [21]),...,etc. Our result provides an alternative approach to those algebras.…”

“…Besides the example above, Theorem 3.5.1 applies to many other interesting algebras. However, if the generating relations of U involve degree 1 part as it did in [2,18,20,21,24], then the situation becomes more complicated. We can not prove the PBW theorem under the hypothesis on B as in Theorem 3.5.1.…”

“…This does not occur coincidently. Shepler and Witherspoon proved in [20] (more recently [21]) the PBW theorem for any skew group algebra obtained from an action of a finite group on a Koszul algebra. More generally, Walton and Witherspoon [24] proved the PBW theorem for any smash product algebra obtained from a Hopf algebra acting on a Koszul algebra.…”