Skip to main content Accessibility help
×
Home

Extensions of tensor categories by finite group fusion categories

  • SONIA NATALE (a1) (a2) and CIEM – CONICET (a1) (a2)

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 × GG 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.

Copyright

Footnotes

Hide All

Partially supported by CONICET and SeCYT–UNC.

Footnotes

References

Hide All
[1] Bruguières, A. and Natale, S.. Exact sequences of tensor categories. Int. Math. Res. Not. (24) (2011), 5644–5705.
[2] Bruguières, A. and Natale, S.. Central exact sequences of tensor categories, equivariantization and applications. J. Math. Soc. Japan 66 (2014), 257287.
[3] Drinfeld, V., Gelaki, S., Nikshych, D. and Ostrik, V.. On braided fusion categories I. Sel. Math. New Ser. 16 (2010), 1119.
[4] Etingof, P. and Gelaki, S.. Exact sequences of tensor categories with respect to a module category. Adv. Math. 308 (2017), 11871208.
[5] Etingof, P., Gelaki, S., Nikshych, D. and Ostrik, V.. Tensor categories. Math. Surv. Monogr. 205 (Amer. Math. Soc., Providence, RI, 2015).
[6] Etingof, P., Nikshych, D. and Ostrik, V.. Weakly group-theoretical and solvable fusion categories. Adv. Math 226 (2011), 176205.
[7] Gelaki, S.. Exact factorisations and extensions of fusion categories. J. Algebra 480 (2017), 505518.
[8] Gelaki, S. and Nikshych, D.. Nilpotent fusion categories. Adv. Math. 217 (2008), 10531071.
[9] Joyal, A. and Street, R.. Braided tensor categories. Adv. Math. 102 (1993), 2078.
[10] Kac, G. I.. Extensions of groups to ring groups. Math. USSR Sbornik 5 (1968), 451474.
[11] Masuoka, A.. Extensions of Hopf algebras. Trab. Mat. 41/99, FaMAF. (Univ. Nac. de Córdoba, 1999).
[12] Mombelli, M. and Natale, S.. Module categories over equivariantised tensor categories. Mosc. Math. J. 17 (1), (2017), 97128.
[13] Montgomery, S. and Witherspoon, S. J.. Irreducible representations of crossed products. J. Pure Appl. Algebra 129 (1998), 315–326.
[14] Müger, M.. Galois extensions of braided tensor categories and braided crossed G-categories. J. Algebra 277 (2004), 256281.
[15] Natale, S.. Crossed actions of matched pairs of groups on tensor categories. Tohoku Math. J . 68 (3) (2016), 377405.
[16] Natale, S.. The core of a weakly group-theoretical fusion category. Int. J. Math. 29, No. 2, Article ID 1850012 (2018), 23 p.
[17] Ostrik, V.. Module categories, weak Hopf algebras and modular invariants. Transform. Groups 8 (2003), 177206.
[18] Pareigis, B.. On braiding and dyslexia. J. Algebra 171 (1995), 413425.
[19] Schauenburg, P.. The monoidal center construction and bimodules. J. Pure Appl. Algebra 158 (2001), 325346.

MSC classification

Extensions of tensor categories by finite group fusion categories

  • SONIA NATALE (a1) (a2) and CIEM – CONICET (a1) (a2)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed