On the trace of the antipode and higher indicators
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. © 2011 Hebrew University Magnes Press.
Publication Source (Journal or Book title)
Israel Journal of Mathematics
Kashina, Y., Montgomery, S., & Ng, S. (2012). On the trace of the antipode and higher indicators. Israel Journal of Mathematics, 188 (1), 57-89. https://doi.org/10.1007/s11856-011-0092-7