Weakly group-theoretical and solvable fusion categories

Academic Article


  • 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.
  • Authors

  • Etingof, Pavel
  • Nikshych, Dmitri
  • Ostrik, Victor
  • Status

    Publication Date

  • January 15, 2011
  • Has Subject Area

    Published In


  • Braided fusion categories
  • Categorical Morita equivalence
  • Fusion categories
  • Group-theoretical fusion categories
  • Semisimple Hopf algebras
  • Solvable fusion categories
  • Digital Object Identifier (doi)

    Start Page

  • 176
  • End Page

  • 205
  • Volume

  • 226
  • Issue

  • 1