Nakayama automorphisms of PBW deformations and Hopf actions

Poincaré-Birkhoff-Witt (PBW) deformations of Artin-Schelter regular algebras are skew Calabi- Yau. We prove that the Nakayama automorphisms of such PBW deformations can be obtained from their homogenizations. Some Calabi-Yau properties are generalized without Koszul assumption. We also show that the Nakayama automorphisms of such PBW deformations control Hopf actions on them.