Weakly Group-Theoretical and Solvable Fusion Categories
Name
Etingof Adv Math 2011.pdf
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399.88 KB
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Author(s)
Etingof, Pavel I.
Nikshych, Dmitri
Ostrik, Viktor
Date Issued
January 2011
Journal
Advances in Mathematics
Publisher
Elsevier
Citation
Etingof, Pavel, Dmitri Nikshych, and Victor Ostrik. “Weakly Group-theoretical and Solvable Fusion Categories.” Advances in Mathematics 226.1 (2011) : 176-205.
Version
Author's final manuscript
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 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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1016/j.aim.2010.06.009
