Vol. 3, No. 8, 2009

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Shlomo Gelaki, Deepak Naidu and Dmitri Nikshych

Vol. 3 (2009), No. 8, 959–990
Abstract

Let $\mathsc{C}$ be a fusion category faithfully graded by a finite group $G$ and let $\mathsc{D}$ be the trivial component of this grading. The center $\mathsc{Z}\left(\mathsc{C}\right)$ of $\mathsc{C}$ is shown to be canonically equivalent to a $G$-equivariantization of the relative center ${\mathsc{Z}}_{\mathsc{D}}\left(\mathsc{C}\right)$. We use this result to obtain a criterion for $\mathsc{C}$ 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
Primary: 16W30
Secondary: 18D10