Faithful simple objects, orders and gradings of fusion categories

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 that have a faithful simple object and show that their universal grading groups 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.