Deformation Theory and Hopf Actions on Koszul Algebras

We consider graded deformations and PBW deformations of algebras defined over noncommutative algebras. We explain how fibers of graded deformations correspond to filtered algebras admitting a PBW property, with focus on smash product algebras for Hopf algebras acting on quadratic algebras. In particular, we describe all PBW deformations arising from Hopf actions on Koszul algebras, giving Hopf-Koszul Hecke algebras. Alexander-Whitney and Eilenberg-Zilber maps for twisted tensor products transfer homological information between resolutions and convert conditions on Hochschild cocycles and Gerstenhaber brackets into explicit PBW conditions.