Centers of graded fusion categories

Gelaki, Shlomo; Naidu, Deepak; Nikshych, Dmitri

doi:10.2140/ant.2009.3.959

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Centers of graded fusion categories

Shlomo Gelaki, Deepak Naidu and Dmitri Nikshych

Vol. 3 (2009), No. 8, 959–990

DOI: 10.2140/ant.2009.3.959

Abstract

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 $Z_{D} (C)$ . 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.

Keywords

fusion categories, braided categories, graded tensor categories

Mathematical Subject Classification 2000

Primary: 16W30

Secondary: 18D10

Milestones

Received: 21 May 2009

Revised: 31 August 2009

Accepted: 9 November 2009

Published: 25 December 2009

Authors

Shlomo Gelaki
Department of Mathematics Technion-Israel Institute of Technology 32000 Haifa Israel

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

Vol. 3, No. 8, 2009