Indicators of Hopf algebras in positive characteristic

TL;DR

This paper investigates the behavior of nth indicators in finite-dimensional Hopf algebras over fields of positive characteristic, revealing a shared sequence pattern under specific structural assumptions.

Contribution

It demonstrates that in positive characteristic, indicators follow a common sequence pattern when the coradical is a local Hopf subalgebra, extending understanding of their properties.

Findings

Indicators in positive characteristic share the same sequence pattern.

The pattern holds when the coradical is a local Hopf subalgebra.

Provides insight into gauge invariance in positive characteristic contexts.

Abstract

The notion of $n$-th indicator for a finite-dimensional Hopf algebra was introduced by Kashina, Montgomery and Ng as gauge invariance of the monoidal category of its representations. The properties of these indicators were further investigated by Shimizu. In this short note, we show that the indicators appearing in positive characteristic all share the same sequence pattern if we assume the coradical of the Hopf algebra is a local Hopf subalgebra.