The Centralizer on H-Separable Smash Products

Let H be a finite dimensional Hopf algebra over a field and A an H-module algebra. The H induces an action on the CA#H(A) by adjoint and CA#H(A)H= Z(A # H) = C,where CA#H(A) denotes the centralizer which algebra A in A # H and Z(A # H) the center of A # H.The aim of this paper is to discuss ,the Galois conditions on the centralizer CA# H(A).We prove that CA# H(A)/ZA # H is H* -Galois if and only if CA# H(A)# H/CA# H(A) is H-separable). Furthermore , if H is a finite dimensional semisimple Hopf algebra and CA# H(A)# H is an Azumaya C-algebra or A # H/A is H-separable, CA# H(A) satisfies the double centralizer property in CA# H(A)# H, CA# H(A)/C is separable and there exists a cocommutative left integral t ∈∫1H,then CA# H(A)/C is H*-Galois.

Let H be a finite dimensional Hopf algebra over a field and A an H-module algebra. The H induces an action on the CA# H( A ) by adjoint and CA# H (A)H=Z(A # H) = C, where CA# H(A) denotes the centralizer which algebra A in A # H and Z(A # H) the center of A # H.The aim of this paper is to discuss , the Galois conditions on the centralizer CA# H(A) . We prove that CA# H( A )/ZA # H is H * -Galois if and only if CA#H(A)# H/CA#H(A) is H-separable). Furthermore , if H is a finite dimensional semisimple Hopf algebra and CA# H(A )# H is an Azumaya C-algebra or A # H/A is H-separable, CA# H( A ) satisfies the double centralizer property in CA# H(A)# H, CA# H( A)/C is separable and there exists a cocommutative left integral t∈∫H1 , then CA#H(A)/C is H*-Galois.