Faithful simple objects, orders and gradings of fusion categories

TL;DR

This paper explores the relationship between simple objects and grading groups in fusion categories, establishing conditions for the existence of faithful simple objects and classifying certain modular categories.

Contribution

It proves that fusion categories with a faithful simple object have cyclic universal grading groups and classifies modular categories with generalized Tambara-Yamagami rules.

Findings

Universal grading group of a faithful simple object is cyclic.

Braided nilpotent fusion categories with cyclic grading have faithful simple objects.

Classification of modular categories with Tambara-Yamagami fusion rules.

Abstract

We establish some relations between the orders of simple objects in a fusion category and the structure of its universal grading group. We consider fusion categories which have a faithful simple object and show that its universal grading group must be cyclic. As for the converse, we prove that a braided nilpotent fusion category with cyclic universal grading group always has a faithful simple object. We study the universal grading of fusion categories with generalized Tambara-Yamagami fusion rules. As an application, we classify modular categories in this class and describe the modularizations of braided Tambara-Yamagami fusion categories.