Higher Frobenius-Schur indicators for semisimple Hopf algebras in positive characteristic

TL;DR

This paper extends the concept of higher Frobenius-Schur indicators to semisimple Hopf algebras over fields of positive characteristic, providing new formulas and invariance properties that mirror the characteristic zero case.

Contribution

It introduces a formula for the antipode squared in positive characteristic, enabling the definition of higher Frobenius-Schur indicators with properties similar to those in characteristic zero.

Findings

Defined higher Frobenius-Schur indicators in positive characteristic

Proved these indicators are gauge invariants of the representation category

Extended properties of indicators from characteristic zero to positive characteristic

Abstract

Let $H$ be a semisimple Hopf algebra over an algebraically closed field $\mathbbm{k}$ of characteristic $p>\dim_{\mathbbm{k}}(H)^{1/2}$. We show that the antipode $S$ of $H$ satisfies the equality $S^2(h)=\mathbf{u}h\mathbf{u}^{-1}$, where $h\in H$, $\mathbf{u}=S(\Lambda_{(2)})\Lambda_{(1)}$ and $\Lambda$ is a nonzero integral of $H$. The formula of $S^2$ enables us to define higher Frobenius-Schur indicators for the Hopf algebra $H$. This generalizes the notions of higher Frobenius-Schur indicators from the case of characteristic 0 to the case of characteristic $p>\dim_{\mathbbm{k}}(H)^{1/2}$. These indicators defined here share some properties with the ones defined over a field of characteristic 0. Especially, all these indicators are gauge invariants for the tensor category Rep$(H)$ of finite dimensional representations of $H$.