Frobenius monoidal functors induced by Frobenius extensions of Hopf algebras

TL;DR

This paper demonstrates that induction along Frobenius extensions of Hopf algebras is a Frobenius monoidal functor, with applications to quantum groups and classifications of subalgebras.

Contribution

It establishes the Frobenius monoidal nature of induction functors in broad contexts and classifies certain unimodular Hopf subalgebras.

Findings

Induction along Frobenius extensions is Frobenius monoidal for all finite-dimensional Hopf algebras.

Induction functors from unimodular Hopf subalgebras to small quantum groups are Frobenius monoidal.

Stronger conditions allow extension to braided Frobenius monoidal functors on Yetter--Drinfeld modules.

Abstract

We show that induction along a Frobenius extension of Hopf algebras is a Frobenius monoidal functor in great generality, in particular, for all finite-dimensional and all pointed Hopf algebras. As an application, we show that induction functors from unimodular Hopf subalgebras to small quantum groups at roots of unity are Frobenius monoidal functors and classify such unimodular Hopf subalgebras. Moreover, we present stronger conditions on Frobenius extensions under which the induction functor extends to a braided Frobenius monoidal functor on categories of Yetter--Drinfeld modules. We show that these stronger conditions hold for any extension of finite-dimensional semisimple and co-semisimple (or, more generally, unimodular and dual unimodular) Hopf algebras.