Frobenius monoidal functors induced by Frobenius extensions of Hopf algebras
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909001841639424 |
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| author | Flake, Johannes Laugwitz, Robert Posur, Sebastian |
| author_facet | Flake, Johannes Laugwitz, Robert Posur, Sebastian |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_15056 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Frobenius monoidal functors induced by Frobenius extensions of Hopf algebras Flake, Johannes Laugwitz, Robert Posur, Sebastian Quantum Algebra Category Theory Representation Theory 16T05 (Primary) 16L60, 18M05, 18M15, 16L60 (Secondary) 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. |
| title | Frobenius monoidal functors induced by Frobenius extensions of Hopf algebras |
| topic | Quantum Algebra Category Theory Representation Theory 16T05 (Primary) 16L60, 18M05, 18M15, 16L60 (Secondary) |
| url | https://arxiv.org/abs/2412.15056 |