On the classification of certain fusion categories

TL;DR

This paper classifies integral fusion categories of dimension pq^2 and a family of Z/3Z-graded fusion categories, advancing the understanding of their structure and examples, especially where not all are group-theoretical.

Contribution

It provides a complete classification of integral fusion categories of specific dimensions and introduces a classification of certain Z/3Z-graded fusion categories, expanding the known landscape.

Findings

Classified integral fusion categories of dimension pq^2

Classified a family of Z/3Z-graded fusion categories

Identified cases where not all categories are group-theoretical

Abstract

We advance the classification of fusion categories in two directions. Firstly, we completely classify integral fusion categories -- and consequently, semi-simple Hopf algebras -- of dimension $pq^2$, where $p$ and $q$ are distinct primes. This case is especially interesting because it is the simplest class of dimensions where not all integral fusion categories are group-theoretical. Secondly, we classify a certain family of $\ZZ/3\ZZ$-graded fusion categories, which are generalizations of the $\ZZ/2\ZZ$-graded Tambara-Yamagami categories. Our proofs are based on the recently developed theory of extensions of fusion categories.