On the trace of the antipode and higher indicators |
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Authors: | Yevgenia Kashina Susan Montgomery Siu-Hung Ng |
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Institution: | 1.Department of Mathematical Sciences,DePaul University,Chicago,USA;2.Department of Mathematics,University of Southern California,Los Angeles,USA;3.Department of Mathematics,Iowa State University,Ames,USA |
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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 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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