On Hopf Galois Extension of Separable Algebras

In this paper, the classical Galois theory to the $H^*$-Galois caseis developed. Let $H$ be a semisimple and cosemisimple Hopf algebraover a field $k$, $A$ a left $H$-module algebra, and $A/A^H$ a right$H^*$-Galois extension. The authors prove that, if $A^H$ is aseparable $k$-algebra, then for any right coideal subalgebra $B$ of$H$, the $B$-invariants $A^B={ain A mid bcdota=varepsilon(b)a, forall bin B}$ is a separable $k$-algebra.They also establish a Galois connection between right coidealsubalgebras of $H$ and separable subalgebras of $A$ containing $A^H$as in the classical case. The results are applied to the case$H=(kG)^*$ for a finite group $G$ to get a Galois 1-1correspondence.