On the trace of the antipode and higher indicators

Details On the trace of the antipode and higher indicators On the trace of the antipode and higher indicators

2009-10-08

arxiv.org/abs/0910.1628v2

Mathematics

Quantum Algebra

additional references, fixed some typos, minor additions including a questions and some remarks

Scientific paper

10.1007/s11856-011-0092-7

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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