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Extensions of tensor categories by finite group fusion categories

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

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

REFERENCES

Bruguières, A. and Natale, S.. Exact sequences of tensor categories. Int. Math. Res. Not. (24) (2011), 5644–5705.CrossRefGoogle Scholar
Bruguières, A. and Natale, S.. Central exact sequences of tensor categories, equivariantization and applications. J. Math. Soc. Japan 66 (2014), 257–287.CrossRefGoogle Scholar
Drinfeld, V., Gelaki, S., Nikshych, D. and Ostrik, V.. On braided fusion categories I. Sel. Math. New Ser. 16 (2010), 1–119.CrossRefGoogle Scholar
Etingof, P. and Gelaki, S.. Exact sequences of tensor categories with respect to a module category. Adv. Math. 308 (2017), 1187–1208.CrossRefGoogle Scholar
Etingof, P., Gelaki, S., Nikshych, D. and Ostrik, V.. Tensor categories. Math. Surv. Monogr. 205 (Amer. Math. Soc., Providence, RI, 2015).CrossRefGoogle Scholar
Etingof, P., Nikshych, D. and Ostrik, V.. Weakly group-theoretical and solvable fusion categories. Adv. Math 226 (2011), 176–205.CrossRefGoogle Scholar
Gelaki, S.. Exact factorisations and extensions of fusion categories. J. Algebra 480 (2017), 505–518.CrossRefGoogle Scholar
Gelaki, S. and Nikshych, D.. Nilpotent fusion categories. Adv. Math. 217 (2008), 1053–1071.CrossRefGoogle Scholar
Joyal, A. and Street, R.. Braided tensor categories. Adv. Math. 102 (1993), 20–78.CrossRefGoogle Scholar
Kac, G. I.. Extensions of groups to ring groups. Math. USSR Sbornik 5 (1968), 451–474.CrossRefGoogle Scholar
Masuoka, A.. Extensions of Hopf algebras. Trab. Mat. 41/99, FaMAF. (Univ. Nac. de Córdoba, 1999).Google Scholar
Mombelli, M. and Natale, S.. Module categories over equivariantised tensor categories. Mosc. Math. J. 17 (1), (2017), 97–128.CrossRefGoogle Scholar
Montgomery, S. and Witherspoon, S. J.. Irreducible representations of crossed products. J. Pure Appl. Algebra 129 (1998), 315–326.CrossRefGoogle Scholar
Müger, M.. Galois extensions of braided tensor categories and braided crossed G-categories. J. Algebra 277 (2004), 256–281.CrossRefGoogle Scholar
Natale, S.. Crossed actions of matched pairs of groups on tensor categories. Tohoku Math. J . 68 (3) (2016), 377–405.CrossRefGoogle Scholar
Natale, S.. The core of a weakly group-theoretical fusion category. Int. J. Math. 29, No. 2, Article ID 1850012 (2018), 23 p.CrossRefGoogle Scholar
Ostrik, V.. Module categories, weak Hopf algebras and modular invariants. Transform. Groups 8 (2003), 177–206.CrossRefGoogle Scholar
Pareigis, B.. On braiding and dyslexia. J. Algebra 171 (1995), 413–425.CrossRefGoogle Scholar
Schauenburg, P.. The monoidal center construction and bimodules. J. Pure Appl. Algebra 158 (2001), 325–346.CrossRefGoogle Scholar