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.
Keywords
fusion category, graded fusion category, faithful object,
universal grading group
Facultad de Matemática, Astronomía y
Física
Universidad Nacional de Córdoba
CIEM-CONICET
Medina Allende s/n
Ciudad Universitaria
5000 Córdoba
Argentina
