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Morales, Yiby; Müller, Monique; Plavnik, Julia; Ros Camacho, Ana; Tabiri, Angela; Walton, Chelsea

Algebraic structures in group-theoretical fusion categories. (English) Zbl 1530.18025

Algebr. Represent. Theory 26, No. 6, 2399-2431 (2023).
An important question in the theory of a fusion category \(\mathcal{C}\) is to classify indecomposable modules of \(\mathcal{C}\). This is equivalent to classify Morita class of indecomposable separable algebras in \(\mathcal{C}.\) But in general we know very little about this. For the pointed fusion categories, which we understood quite well, this classification is known due to V. Ostrik [Int. Math. Res. Not. 2003, No. 27, 1507–1520 (2003; Zbl 1044.18005)] and S. Natale [SIGMA, Symmetry Integrability Geom. Methods Appl. 13, Paper 042, 9 p. (2017; Zbl 1437.18011)] for a quite long time (one can find this result in [P. Etingof et al., Tensor categories. Providence, RI: American Mathematical Society (AMS) (2015; Zbl 1365.18001)]). This paper generalize this classification the case of group-theoretical fusion categories. The result is interesting and important. Maybe, it is also interesting to consider the setting of weakly group-theoretical fusion categories.
Reviewer: Liu Gongxiang (Nanjing)

MSC:

18M20 Fusion categories, modular tensor categories, modular functors
16D90 Module categories in associative algebras
16H05 Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.)

Keywords:

free functor; Frobenius algebra; Frobenius monoidal functor; group-theoretical fusion category; Morita equivalence; pointed fusion category; separable algebra

Citations:

Zbl 1044.18005; Zbl 1437.18011; Zbl 1365.18001
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References:

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