On the classification of certain fusion categories
David Jordan, Eric Larson
TL;DR
The paper advances the classification of fusion categories in two directions: it completely classifies integral fusion categories of dimension $pq^2$ (distinct primes $p,q$), identifying a new non-group-theoretical family that occurs only when $p$ is odd and divides $q+1$, and proves all dimension $pq^2$ semisimple Hopf algebras are group-theoretical. It also classifies a family of $\\\\mathbb{Z}/3\mathbb{Z}$-graded categorifications of rings $R_{3,A}$ for abelian $A$ with $|A|$ not divisible by $3$, giving detailed parameterizations in terms of $H^3$ data and nondegenerate maps $A\to A^*$. The proofs leverage the theory of extensions of fusion categories, Morita theory, and linear algebra over finite fields to control Lagrangian subspaces and grading structures. Together, these results illuminate the landscape of low-rank fusion categories and their connections to Hopf algebras, with explicit counts of inequivalent categorifications and clear pathways for extending the analysis to higher dimensions.
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.