Non-trivially graded self-dual fusion categories of rank $4$

Abstract

Let $\mathcal{C}$ be a self-dual spherical fusion categories of rank $4$ with non-trivial grading. We complete the classification of Grothendieck ring $K(\mathcal{C})$ of $\mathcal{C}$; that is, we prove that $K(\mathcal{C})\cong Fib\otimes \mathbb{Z}[{\mathbb{Z}}_2]$, where $Fib$ is the Fibonacci fusion ring and $\mathbb{Z}[{\mathbb{Z}}_2]$ is the group ring on ${\mathbb{Z}}_2$. In particular, if $\mathcal{C}$ is braided then it is equivalent to $\text{Fib}⊠{\text{Vec}}_{{\mathbb{Z}}_2}^{\omega }$ as fusion categories, where $\text{Fib}$ is a Fibonacci category and ${\text{Vec}}_{{\mathbb{Z}}_2}^{\omega }$ is a rank $2$ pointed fusion category.

13 pages, rewrite most parts of our paper, to appear in Acta Mathematica Sinica, English Series

References in corpus (1)

• Pivotal fusion categories of rank 3 (with an Appendix written jointly with Dmitri Nikshych)