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Extensions of tensor categories by finite group fusion categories

Part of: Categories with structure Hopf algebras, quantum groups and related topics

Published online by Cambridge University Press:  11 November 2019

SONIA NATALE  and
CIEM – CONICET
SONIA NATALE
Affiliation:
Facultad de Matemática, Astronomía, Física y Computación Universidad Nacional de Córdoba Ciudad Universitaria (5000) Córdoba, Argentina e-mail: [email protected]
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Abstract

We study exact sequences of finite tensor categories of the form Rep G → 𝒞 → 𝒟, where G is a finite group. We show that, under suitable assumptions, there exists a group Γ and mutual actions by permutations ⊳ : Γ × G → G and ⊲ : Γ × G→ Γ that make (G, Γ) into matched pair of groups endowed with a natural crossed action on 𝒟 such that 𝒞 is equivalent to a certain associated crossed extension 𝒟(G,Γ) of 𝒟. Dually, we show that an exact sequence of finite tensor categories VecG → 𝒞 → 𝒟 induces an Aut(G)-grading on 𝒞 whose neutral homogeneous component is a (Z(G), Γ)-crossed extension of a tensor subcategory of 𝒟. As an application we prove that such extensions 𝒞 of 𝒟 are weakly group-theoretical fusion categories if and only if 𝒟 is a weakly group-theoretical fusion category. In particular, we conclude that every semisolvable semisimple Hopf algebra is weakly group-theoretical.

MSC classification

Primary: 18D10: Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories
Secondary: 16T05: Hopf algebras and their applications
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 170 , Issue 1 , January 2021 , pp. 161 - 189
Copyright
© Cambridge Philosophical Society 2019

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Footnotes

†

Partially supported by CONICET and SeCY-UNC.

References

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