On Total Frobenius-Schur Indicators
Abstract
We define total Frobenius-Schur indicator for each object in a spherical fusion category as a certain canonical sum of its higher indicators. The total indicators are invariants of spherical fusion categories. If is the representation category of a semisimple quasi-Hopf algebra , we prove that the total indicators are non-negative integers which satisfy a certain divisibility condition. In addition, if is a Hopf algebra, then all the total indicators are positive. Consequently, the positivity of total indicators is a necessary condition for a quasi-Hopf algebra being gauge equivalent to a Hopf algebra. Certain twisted quantum doubles of finite groups and some examples of Tambara-Yamagami categories are discussed for the sufficiency of this positivity condition.
Keywords
Cite
@article{arxiv.1212.1435,
title = {On Total Frobenius-Schur Indicators},
author = {Gongxiang Liu and Siu-Hung Ng},
journal= {arXiv preprint arXiv:1212.1435},
year = {2015}
}
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20pages Latex