On the classification of certain fusion categories

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 $p{q}^2$, 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 $Z/3Z$-graded fusion categories, which are generalizations of the $Z/2Z$-graded Tambara–Yamagami categories. Our proofs are based on the recently developed theory of extensions of fusion categories.