Hopf algebra actions on differential graded algebras and applications

Abstract: Let $H$ be a finite dimensional semisimple Hopf algebra, $A$ a differential graded (dg for short) $H$-module algebra. Then the smash product algebra $A#H$ is a dg algebra. For any dg $A#H$-module $M$, there is a quasi-isomorphism of dg algebras: $\mathrm{RHom}A(M,M)#H\longrightarrow \mathrm{RHom}{A#H}(M\ot H,M\ot H)$. This result is applied to $d$-Koszul algebras, Calabi-Yau algebras and AS-Gorenstein dg algebras