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 $\ZZ/3\ZZ$-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.

References in corpus (4)

• Group-theoretical properties of nilpotent modular categories
• Module categories over the Drinfeld double of a finite group
• Weakly group-theoretical and solvable fusion categories
• Classification of fusion categories of dimension pq

Cited by in corpus (11)

• The balanced tensor product of module categories
• Fusion 2-categories with no line operators are grouplike
• The Moonshine Anomaly
• Realizing triality and $p$-ality by lattice twisted gauging in (1+1)d quantum spin systems
• Weakly group-theoretical and solvable fusion categories
• A finiteness property for braided fusion categories
• A class of prime fusion categories of dimension $2^N$
• Classification of integral modular data up to rank 13
• Group-theoretical property of non-degenerate fusion categories of FP-dimension $p^2{q}^3$ and $p^3{q}^3$
• Quadratic $d$-numbers
• Fusion rings arising from normal Hopf subalgebras