The commutator Hopf Galois extensions

Abstract

Let \(H\) be a finite dimentional Hopf algebra over a field \(k\) and \(H^*\) the dual Hopf algebra of \(H\). Then a commutator right \(H^*\)-Galois extension \(B\) of \(B^H\) is characterized in terms of the smash product \(B \ne H\) and some relationships between such a \(B\) and the Hopf Galois Azumaya or Hopf Galois Hirata extensions are also given.