Braided extensions of a rank $2$ fusion category

TL;DR

This paper classifies braided extensions of rank 2 fusion categories, revealing they are mostly tensor products of known categories, with some exceptions involving slightly degenerate cases generated by a specific simple object.

Contribution

It provides a complete classification of braided extensions of rank 2 fusion categories, including their fusion rules, grading groups, and dimensions, highlighting new degenerate cases.

Findings

Most braided extensions are tensor products of known categories.

Identifies a new class of slightly degenerate categories generated by a $\

Provides explicit fusion rules and structural properties of these categories.

Abstract

We classify braided extensions $C$ of a rank $2$ fusion category. The result shows that $C$ is tensor equivalent to a Deligne's tensor product of some known categories, except $C$ is slightly degenerate and generated by a $\sqrt{2}$-dimensional simple object. To start with, we describe the fusion rules, universal grading group, and the Frobenius-Perron dimensions of simple objects of $C$ without the restriction that $C$ is braided.