Vol. 303, No. 1, 2019

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Frobenius–Schur indicators for near-group and Haagerup–Izumi fusion categories

Henry Tucker

Vol. 303 (2019), No. 1, 337–359
Abstract

Ng and Schauenburg generalized higher Frobenius–Schur indicators to pivotal fusion categories and showed that these indicators may be computed utilizing the modular data of the Drinfel’d center of the given category. We consider two classes of fusion categories generated by a single noninvertible simple object: near groups, those fusion categories with one noninvertible object, and Haagerup–Izumi categories, those with one noninvertible object for every invertible object. Examples of both types arise as representations of finite or quantum groups or as Jones standard invariants of finite-depth Murray–von Neumann subfactors. We utilize the computations of the tube algebras due to Izumi and to Evans and Gannon to obtain formulae for the Frobenius–Schur indicators of objects in both of these families.

Dedicated to Susan Montgomery

Keywords
tensor category, fusion rules, Frobenius–Schur indicator, Drinfel'd center, modular data, Haagerup subfactor, Hopf algebras
Mathematical Subject Classification 2010
Primary: 18D10, 16T05
Secondary: 46L37
Milestones
Received: 9 June 2017
Revised: 20 January 2019
Accepted: 29 January 2019
Published: 21 December 2019
Authors
Henry Tucker
Department of Mathematics
University of California, San Diego
La Jolla, CA
United States