  Braverman and Gaitsgory gave necessary and sufficient conditions for a
nonhomogeneous quadratic algebra to satisfy the Poincare-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 Poincare-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.
