Weakly Group-Theoretical and Solvable Fusion Categories

[摘要] 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 View the MathML source has the strong Frobenius property (i.e., the dimension of any simple object in an indecomposable View the MathML source-module category divides the dimension of View the MathML source), 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 pq2, where p,q,r are distinct primes.

[发布机构] Elsevier