On the classification of certain fusion categories

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, 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.