The Tannakian radical and the mantle of a braided fusion category

TL;DR

This paper introduces the Tannakian radical and mantle of a braided fusion category, providing a new framework for understanding their structure and classification, with implications for the study of reductive categories.

Contribution

It defines the Tannakian radical and mantle, and establishes their properties, including a canonical central extension as a complete invariant, advancing the classification of braided fusion categories.

Findings

The mantle admits a canonical central extension.

Reductive categories have trivial Tannakian radical.

Several classification results for reductive categories.

Abstract

We define the Tannakian radical of a braided fusion category $\mathcal{C}$ as the intersection of its maximal Tannakian subcategories. The localization of $\mathcal{C}$ corresponding to the Tannakian radical, termed the mantle of $\mathcal{C}$, admits a canonical central extension that serves as a complete invariant of $\mathcal{C}$. The mantle has a trivial Tannakian radical, and we refer to braided fusion categories with this property as reductive. We investigate the properties and structure of reductive categories and prove several classification results.