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

Abstract

Abstract. Braverman and Gaitsgory 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.

Spectral sequences for the cohomology rings of a smash product

“…Our resolution K#L ↑ is a member of their class of resolutions (up to isomorphism). The reader should be aware that the construction given in [16] is somewhat different than the one given here.…”

“…In [7], Guccione and Guccione provide a resolution X of the smash A#Γ which is the tensor product of the bar resolution of A with the bar resolution of Γ, along with some explicit differential. In the case that Γ is a group algebra, Shepler and Witherspoon have provided a class of resolutions of the smash product [16,Section 4]. Our resolution K#L ↑ is a member of their class of resolutions (up to isomorphism).…”

PBW Deformations of Skew Group Algebras in Positive Characteristic

“…The resolution Xr of A = S(V )#G we describe next is from [27], a generalization of a resolution given by Guccione, Guccione, and Valqui [13, §4.1]. It arises as a tensor product of the Koszul resolution of S(V ) with the bar resolution of kG, and we recall only the outcome of this construction here.…”

“…It arises as a tensor product of the Koszul resolution of S(V ) with the bar resolution of kG, and we recall only the outcome of this construction here. For details, see [27]. In the case that the characteristic of k does not divide the order of G, one may simply use the Koszul resolution of S(V ) to obtain homological information about deformations, as in that case the Hochschild cohomology of S(V )#G is isomorphic to the G-invariants in the Hochschild cohomology of S(V ) with coefficients in S(V )#G. See, e.g., [23].…”

“…It is proven in [27], in a more general context, that Xr is a free resolution of the A e -module A. We will need to extend some results on chain maps found in [27]. In the following lemma, we let A ⊗( r +2) denote the bar resolution of A.…”