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arXiv · 0910.1628

On the trace of the antipode and higher indicators

Yevgenia Kashina·Susan Montgomery·Siu-Hung Ng
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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 dim H. 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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Yevgenia Kashina, Susan Montgomery, Siu-Hung Ng. 2009-10-28. On the trace of the antipode and higher indicators. https://doi.org/10.1007/s11856-011-0092-7

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