On Hopf Galois Extension of Separable Algebras

In this paper, the classical Galois theory to the $H^*$-Galois case is developed. Let $H$ be a semisimple and cosemisimple Hopf algebra over 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 a separable $k$-algebra, then for any right coideal subalgebra $B$ of $H$, the $B$-invariants $A^B=\{a\in A \mid b\cdot a=\varepsilon(b)a,\ \forall b\in B\}$ is a separable $k$-algebra. They also establish a Galois connection between right coideal subalgebras 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 1-1 correspondence.