Weakly group-theoretical and solvable fusion categories
arXiv:0809.3031
Abstract
We introduce two new classes of fusion categories which are obtained by a certain procedure from finite groups - weakly group-theoretical categories and solvable categories. These are fusion categories that are Morita equivalent to iterated extensions (in the world of fusion categories) of arbitrary, respectively solvable finite groups. Weakly group-theoretical categories have integer dimension, and all known fusion categories of integer dimension are weakly group theoretical. Our main results are that a weakly group-theoretical category C has the strong Frobenius property (i.e., the dimension of any simple object in an indecomposable C-module category divides the dimension of C), and that any fusion category whose dimension has at most two prime divisors is solvable (a categorical analog of Burnside's theorem for finite groups). This has powerful applications to classification of fusion categories and semsisimple Hopf algebras of a given dimension. In particular, we show that any fusion category of integer dimension <84 is weakly group-theoretical (i.e. comes from finite group theory), and give a full classification of semisimple Hopf algebras of dimensions pqr and pq^2, where p,q,r are distinct primes.
28 pages, latex; added many references and details in proofs
References in corpus (5)
Cited by in corpus (10)
- Classification of Rank 5 Premodular Categories
- Cyclotomic integers, fusion categories, and subfactors
- On the classification of certain fusion categories
- On braided fusion categories I
- On Hopf algebras of dimension 4p
- The small quantum group as a quantum double
- Reductions of tensor categories modulo primes
- Semisimple Hopf algebras of dimension 60
- Module categories over pointed Hopf algebras
- Fusion rings arising from normal Hopf subalgebras