Integral almost square-free modular categories

Abstract

We study integral almost square-free modular categories; i.e., integral modular categories of Frobenius-Perron dimension $p^{n}m$, where $p$ is a prime number, $m$ is a square-free natural number and $\gcd (p,m)=1$. We prove that if $n\le 5$ or $m$ is prime with $m<p$ then they are group-theoretical. This generalizes several results in the literature and gives a partial answer to the question posed by the first author and H. Tucker. As an application, we prove that an integral modular category whose Frobenius-Perron dimensions is odd and less than $1125$ is group-theoretical.

Some typos are corrected. Section 4 in previous version is moved. The current version is revised by Libin Li and Li Dai

References in corpus (2)

• Frobenius property for fusion categories of small integral dimension
• On semisimple Hopf algebras of dimension $2q^3$