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Abstract. We consider when extensions S ⊂ R of subalgebras of a Hopf algebra are β-Frobenius, that is Frobenius of the second kind. Given a Hopf algebra H, we show that when S ⊂ R are Hopf algebras in the Yetter-Drinfeld category for H, the extension is β-Frobenius provided R is finite over S and the extension of biproducts S H ⊂ R H is cleft.More generally we give conditions for an extension to be β-Frobenius; in particular we study extensions of integral type, and consider when the Frobenius property is inherited by the subalgebras of coinvariants.We apply our results to extensions of enveloping algebras of Lie coloralgebras, thus extending a result of Bell and Farnsteiner for Lie superalgebras.

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