On the trace of the antipode and higher indicators

TL;DR

This paper introduces gauge invariants for finite-dimensional Hopf algebras, linking antipode traces and higher Frobenius-Schur indicators to algebraic properties and module categories, with implications for semisimple and quasi-Hopf structures.

Contribution

It defines new gauge invariants for Hopf algebras and explores their properties, especially in the context of semisimple and quasi-Hopf algebras, including positivity and divisibility results.

Findings

Higher indicators are non-negative in modular module categories.

The p-th indicator equals 1 iff p divides the dimension of H.

Existence of a self-dual simple module determined by the second indicator.

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.