Group-theoretical Fusion Categories Research Articles

Reductions of Tensor Categories Modulo Primes

We study good (i.e., semisimple) reductions of semisimple rigid tensor categories modulo primes. A prime p is called good for a semisimple rigid tensor category 𝒞 if such a reduction exists (otherwise, it is called bad). It is clear that a good prime must be relatively prime to the Müger squared norm |V|2 of any simple object V of 𝒞. We show, using the Ito–Michler theorem in finite group theory, that for group-theoretical fusion categories, the converse is true. While the converse is false for general fusion categories, we obtain results about good and bad primes for many known fusion categories (e.g., for Verlinde categories). We also state some questions and conjectures regarding good and bad primes.

Lagrangian Subcategories and Braided Tensor Equivalences of Twisted Quantum Doubles of Finite Groups

We classify Lagrangian subcategories of the representation category of a twisted quantum double D ω (G), where G is a finite group and ω is a 3-cocycle on it. In view of results of [DGNO] this gives a complete description of all braided tensor equivalent pairs of twisted quantum doubles of finite groups. We also establish a canonical bijection between Lagrangian subcategories of Rep(D ω (G)) and module categories over the category $${\rm Vec}_{G} ^{\omega}$$ of twisted G-graded vector spaces such that the dual tensor category is pointed. This can be viewed as a quantum version of V. Drinfeld’s characterization of homogeneous spaces of a Poisson-Lie group in terms of Lagrangian subalgebras of the double of its Lie bialgebra [D]. As a consequence, we obtain that two group-theoretical fusion categories are weakly Morita equivalent if and only if their centers are equivalent as braided tensor categories.

On the exponent of tensor categories coming from finite groups

We describe the exponent of a group-theoretical fusion category C = C(G, ω, F, α) associated to a finite group G in terms of group cohomology. We show that the exponent of C divides both e(ω)expG and (expG)2, where e(ω) is the cohomological order of the 3-cocycle ω. In particular, expC divides (dim C)2.

Frobenius–Schur indicators for a class of fusion categories

We give an explicit description of group-theoretical quasi-Hopf algebras up to gauge equivalence. We then use it to compute the Frobenius-Schur indicators for group-theoretical fusion categories.