On an analogue of a Brauer theorem for fusion categories

TL;DR

This paper extends Brauer's theorem to fusion categories by analyzing faithful objects, revealing that the index of a faithful simple object equals the order of the universal grading group, thus linking algebraic properties.

Contribution

It introduces an analogue of Brauer's theorem for fusion categories and establishes the relationship between the index of faithful simple objects and the universal grading group.

Findings

The index of a faithful simple object equals the order of the universal grading group.

Discusses notions of order and index for faithful objects in fusion categories.

Provides a new perspective on the structure of fusion categories through faithful objects.

Abstract

In this paper we prove an analogue of Brauer's theorem for faithful objects in fusion categories. Other notions, such as the order and the index associated to faithful objects of fusion categories are also discussed. We show that the index of a faithful simple object of a fusion categories coincides with the order of the universal grading group of the fusion category.