# The core of a weakly group-theoretical braided fusion category

## TL;DR

This paper characterizes the core structure of weakly group-theoretical braided fusion categories, showing it decomposes into simpler components, and applies this to classify certain integral modular categories as group-theoretical.

## Contribution

It provides a decomposition theorem for the core of weakly group-theoretical braided fusion categories and characterizes their solvability and group-theoretical nature.

## Key findings

- Core decomposes into a tensor product of pointed and Ising categories
- Integral categories have pointed weakly anisotropic cores
- Integral modular categories with simple objects of dimension ≤2 are group-theoretical

## Abstract

We show that the core of a weakly group-theoretical braided fusion category $\C$ is equivalent as a braided fusion category to a tensor product $\B \boxtimes \D$, where $\D$ is a pointed weakly anisotropic braided fusion category, and $\B \cong \vect$ or $\B$ is an Ising braided category. In particular, if $\C$ is integral, then its core is a pointed weakly anisotropic braided fusion category. As an application we give a characterization of the solvability of a weakly group-theoretical braided fusion category. We also prove that an integral modular category all of whose simple objects have Frobenius-Perron dimension at most 2 is necessarily group-theoretical.