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

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.