On the trace of the antipode and higher indicators

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.

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

References in corpus (1)

• Central Invariants and Frobenius-Schur Indicators for Semisimple Quasi-Hopf Algebras

Cited by in corpus (8)

• The characteristic polynomial of the Adams operators on graded connected Hopf algebras
• Higher Gauss sums of modular categories
• The pivotal cover and Frobenius-Schur indicators
• Gauge invariants from the powers of antipodes
• On Two Invariants of Three Manifolds from Hopf Algebras
• Indicators of Tambara-Yamagami categories and Gauss sums
• Computing indicators of Radford algebras
• Polyadic Hopf algebras and quantum groups