Frobenius extensions of subalgebras of Hopf algebras

Abstract

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.

The relative modular object and Frobenius extensions of finite Hopf algebras

“…Frobenius-type properties of extensions of Hopf algebras and of related algebras have been studied extensively; see, e.g., [Sch92, Doi97, FMS97, Kad99, KS01, KS02]. Fischman, Montgomery and Schneider [FMS97] showed that the Frobenius property of an extension of finite-dimensional Hopf algebras is controlled by their modular functions. The aim of this paper is to formulate and prove a generalization of their result in the setting of finite tensor categories, a class of tensor categories including the representation category of a finite-dimensional Hopf algebra.…”

Homological Properties of Color Lie Superalgebras

“…, θ n ] we have gldim gr(U (L + )) = n = dim L + . By[6, Corollary 6.3] Λ is a Frobenius algebra; hence injdim gr(U) gr(U ) =n = dim L + as gr(U ) ∼ = Λ γ [θ 1 , θ 2 , . .…”

“…Then U (L) is Auslander-Gorenstein and Cohen-Macaulay and thus has a quasi-Frobenius classical quotient ring.Proof. We have Λ is Auslander-Gorenstein and Cohen-Macaulay since Λ is a Frobenius algebra (see[6, Corollary 6.3]). As gr(U (L)) is an iterated Ore extension of Λ where each iteration is of the form R[x; σ], it follows from [10, Lemma (ii), p. 184] and[5, Theorem 4.2] that gr(U (L)) is Auslander-Gorenstein and Cohen-Macaulay.…”

Support varieties and representation type of self-injective algebras

“…According to Lemma 5.3 and the proof of [23,Theorem 4.3], we obtain cx u >0 q (g) (C) = cx u ≥0 q (g) (C) ≥ 3 unless r = 1. Since u >0 q (g) is a Yetter-Drinfeld Hopf algebra over the group algebra C[(Z/ℓZ) r ], u >0 q (g) is a Frobenius algebra (see [24,Corollary 5.8] or [43, Proposition 2.10(3)]) and therefore self-injective. Thus, if r ≥ 2, Theorem 2.1 in conjunction with Corollary 5.2 implies that u >0 q (g) is wild.…”

“…By the Krull-Remak-Schmidt Theorem, R ∼ = R * as R-modules, and so R is Frobenius. (For a proof in a much more general context, see [24,Corollary 5.8] or [43,Proposition 2.10(3)]. ) Since R is Frobenius, it is self-injective and the result now follows from Theorem 2.1 by choosing the block in which an indecomposable summand of M of complexity at least 3 lies.…”