Hopf algebra actions on differential graded algebras and applications

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