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Faithful simple objects, orders and gradings of fusion categories

Sonia Natale

Algebraic & Geometric Topology 13 (2013) 1489–1511

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.

fusion category, graded fusion category, faithful object, universal grading group
Mathematical Subject Classification 2010
Primary: 18D10, 16T05
Received: 7 October 2011
Revised: 29 October 2012
Accepted: 22 December 2012
Published: 30 April 2013
Sonia Natale
Facultad de Matemática, Astronomía y Física
Universidad Nacional de Córdoba
Medina Allende s/n
Ciudad Universitaria
5000 Córdoba