On the trace of the antipode and higher indicators

Understand

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.