On Hopf Galois extension of separable algebras

Abstract

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 ∈ A | b · a = ε(b)a, ∀b ∈ 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.

Keywords

Semisimple Hopf algebra / Hopf Galois extension / Separable algebra / Galois connection

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