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 pq2, 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.