A note on frobenius extensions in hopf algebras

Abstract

Let H be a Hopf algebra over a field with bijective antipode and K a Hopf subalgebra with K =^coπ(H)(H), where π(H)= H/HK⁺.Using Miyashita-Ulbrich type action on Endl^−H _−K(H), we show (Cor of Thm 2) that the extension K ⊂H is of right integral type in the sense of [FMS] if and only if it is β-Frobenius such that both βand the β-Frobenius map are H-colinear. We also improve some results in [FMS].

Throughout H denotes a Hopf algebra over a field k with comultiplication △, counit εand antipode S. Let B ⊂A be an extension of right H-comodule algebras, that is, A is a right H-comodule algebra with structure map ρ_A and B is a subalgebra with ρ_A(B) ⊂ B⊗H

In this note we consider when B ⊂A is a β-Frobenius extension such that both βand the corresponding β-Frobenius map are H-colinear. Following [T] we call such an extension a β-H-Frobenius extension, we follow Fischman-Montgomery-Schneider's paper [FMS] for the terminology on β-Frobenius extensions, extensions of right integral type, etc.