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

TL;DR

This paper classifies non-simple modular fusion categories with global dimension p^2, showing they are Grothendieck equivalent to specific categories related to sl_2 at certain levels, based on algebraic number properties.

Contribution

It provides a classification of non-simple modular fusion categories of dimension p^2, linking them to categories derived from sl_2 and algebraic units, expanding understanding of fusion category structures.

Findings

Classification of non-simple modular fusion categories of dimension p^2

Identification of Grothendieck equivalence to specific sl_2-based categories

Connection to algebraic units and Tannakian subcategories

Abstract

Let $p\geq5$ 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 $p^2$.