Frobenius property for fusion categories of small integral dimension

TL;DR

This paper proves that fusion categories with certain small integral dimensions are of Frobenius type, extending the classification of such categories and confirming their Frobenius property for dimensions less than 120.

Contribution

It establishes the Frobenius property for fusion categories of dimensions 84 and 90, and extends this to all weakly integral categories below 120.

Findings

Fusion categories of dimension 84 are of Frobenius type.

Fusion categories of dimension 90 are of Frobenius type.

All weakly integral fusion categories with dimension less than 120 are of Frobenius type.

Abstract

Let k be an algebraically closed field of characteristic zero. In this paper we prove that fusion categories of Frobenius-Perron dimensions 84 and 90 are of Frobenius type. Combining this with previous results in the literature, we obtain that every weakly integral fusion category of Frobenius-Perron dimension less than 120 is of Frobenius type.