Vol. 3, No. 8, 2009

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Centers of graded fusion categories

Shlomo Gelaki, Deepak Naidu and Dmitri Nikshych

Vol. 3 (2009), No. 8, 959–990

Let C be a fusion category faithfully graded by a finite group G and let D be the trivial component of this grading. The center Z(C) of C is shown to be canonically equivalent to a G-equivariantization of the relative center ZD(C). We use this result to obtain a criterion for C to be group-theoretical and apply it to Tambara–Yamagami fusion categories. We also find several new series of modular categories by analyzing the centers of Tambara–Yamagami categories. Finally, we prove a general result about the existence of zeroes in S-matrices of weakly integral modular categories.

fusion categories, braided categories, graded tensor categories
Mathematical Subject Classification 2000
Primary: 16W30
Secondary: 18D10
Received: 21 May 2009
Revised: 31 August 2009
Accepted: 9 November 2009
Published: 25 December 2009
Shlomo Gelaki
Department of Mathematics
Technion-Israel Institute of Technology
32000 Haifa
Deepak Naidu
Department of Mathematics
Texas A&M University
College Station, TX 77843
United States
Dmitri Nikshych
Department of Mathematics and Statistics
University of New Hampshire
Durham, NH 03824
United States