On integral fusion categories with low-dimensional simple objects

TL;DR

This paper investigates integral fusion categories over an algebraically closed field of characteristic zero with simple objects of Frobenius-Perron dimension at most 3, proving they are of Frobenius type and not simple.

Contribution

It establishes that such fusion categories are of Frobenius type and cannot be simple, advancing understanding of low-dimensional fusion categories.

Findings

Fusion categories with simple objects of FP dimension ≤ 3 are of Frobenius type.

Such categories are proven to be non-simple.

The results contribute to classification efforts of low-dimensional fusion categories.

Abstract

Let $k$ be an algebraically closedfield of characteristic zero. In this paper we consider an integral fusion category over $k$ in which the Frobenius-Perron dimensions of its simple objects are at most 3. We prove that such fusion category is of Frobenius type. In addition, we also prove that such fusion category is not simple.