Classification of certain weakly integral fusion categories

TL;DR

This paper classifies weakly integral fusion categories of small Frobenius-Perron dimension, showing they are either solvable or group-theoretical, thereby completing their Morita equivalence classification.

Contribution

It proves that certain classes of braided fusion categories are weakly group-theoretical and completes the classification of weakly integral fusion categories under specific dimension constraints.

Findings

Fusion categories of dimension less than 120 are either solvable or group-theoretical.

Braided fusion categories of certain dimensions are weakly group-theoretical.

Fusion categories of dimensions 84 and 90 are either solvable or group-theoretical.

Abstract

We prove that braided fusion categories of Frobenius-Perron $p^mq^nd$ or $p^2q^2r^2$ 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, we obtain 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, we prove that fusion categories (not necessarily braided) of Frobenius-Perron dimension $84$ and $90$ 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 we complete the classification of all these fusion categories in terms of Morita equivalence.