Classification of Certain Weakly Integral Fusion Categories

Abstract

The author proves that braided fusion categories of Frobenius-Perron dimension p^mqⁿd or p²q²r² are weakly group-theoretical, where p, q, r are distinct prime numbers, d is a square-free natural number such that (pq, d) = 1. As an application, the author obtains that weakly integral braided fusion categories of Frobenius-Perron dimension less than 1800 are weakly group-theoretical, and weakly integral braided fusion categories of odd dimension less than 33075 are solvable. For the general case, the author proves that fusion categories (not necessarily braided) of Frobenius-Perron dimension 84 and 90 are either solvable or group-theoretical. Together with the results in the literature, this shows that every weakly integral fusion category of Frobenius-Perron dimension less than 120 is either solvable or group-theoretical. Thus the author completes the classification of all these fusion categories in terms of Morita equivalence.

Keywords

Solvable fusion categories / Group-theoretical fusion categories / Weakly group-theoretical fusion categories / Frobenius property / 18M20 / 18M15

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