Pre-modular fusion categories of global dimensions $p^2$

Abstract

Let $p\ge 5$ be a prime, we show that a non-pointed modular fusion category $\mathcal{C}$ is Grothendieck equivalent to $\mathcal{C}({\mathfrak{sl}}_2,2(p-1){)}_A^0$ if and only if $\dim (\mathcal{C})=p\cdot u$, where $u$ is a certain totally positive algebraic unit and $A$ is the regular algebra of the Tannakian subcategory $\text{Rep}({\mathbb{Z}}_2)\subseteq \mathcal{C}({\mathfrak{sl}}_2,2(p-1))$. As a direct corollary, we classify non-simple modular fusion categories of global dimensions $p^2$.

25 pages; comments are welcome