Abstract
We introduce two kinds of gauge invariants for any finite-dimensional Hopf algebra H. When H is semisimple over C, these invariants are, respectively, the trace of the map induced by the antipode on the endomorphism ring of a self-dual simple module, and the higher Frobenius-Schur indicators of the regular representation. We further study the values of these higher indicators in the context of complex semisimple quasi-Hopf algebras H. We prove that these indicators are non-negative provided the module category over H is modular, and that for a prime p, the p-th indicator is equal to 1 if, and only if, p is a factor of dimH. As an application, we show the existence of a non-trivial self-dual simple H-module with bounded dimension which is determined by the value of the second indicator.
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Dedicated to Hans-Jürgen Schneider on the occasion of his 65th birthday
This author was supported by a grant from the Faculty Research and Development Program of the College of Liberal Arts and Sciences at DePaul University.
This author was supported by NSF grant DMS 07-01291.
This author was supported by NSA grant H98230-08-1-0078.
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Kashina, Y., Montgomery, S. & Ng, SH. On the trace of the antipode and higher indicators. Isr. J. Math. 188, 57–89 (2012). https://doi.org/10.1007/s11856-011-0092-7
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DOI: https://doi.org/10.1007/s11856-011-0092-7