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Abstract: We introduce two new classes of fusion categories which are obtained by acertain procedure from finite groups - weakly group-theoretical categories andsolvable categories. These are fusion categories that are Morita equivalent toiterated extensions in the world of fusion categories of arbitrary,respectively solvable finite groups. Weakly group-theoretical categories haveinteger dimension, and all known fusion categories of integer dimension areweakly group theoretical. Our main results are that a weakly group-theoreticalcategory C has the strong Frobenius property i.e., the dimension of any simpleobject in an indecomposable C-module category divides the dimension of C, andthat any fusion category whose dimension has at most two prime divisors issolvable a categorical analog of Burnside-s theorem for finite groups. Thishas powerful applications to classification of fusion categories andsemsisimple Hopf algebras of a given dimension. In particular, we show that anyfusion category of integer dimension <84 is weakly group-theoretical i.e.comes from finite group theory, and give a full classification of semisimpleHopf algebras of dimensions pqr and pq^2, where p,q,r are distinct primes.

Autor: Pavel Etingof, Dmitri Nikshych, Victor Ostrik

Fuente: https://arxiv.org/

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