A note on the order of the antipode of a pointed Hopf algebra

TL;DR

This paper investigates the order of the antipode in pointed Hopf algebras, introducing an invariant that links to the antipode's order, with results differing based on the characteristic of the base field.

Contribution

It defines a new invariant for pointed Hopf algebras and establishes its connection to the antipode's order, extending previous results to infinite-dimensional cases.

Findings

In characteristic zero, antipode order is either 1 or 2m_H.

In coradically graded cases, antipode order finiteness is equivalent to m_H being finite.

In positive characteristic, results are generalized to infinite-dimensional Hopf algebras.

Abstract

Let $k$ be a field and let $H$ denote a pointed Hopf $k$-algebra with antipode $S$. We are interested in determining the order of $S$. Building on the work done by Taft and Wilson $[7]$, we define an invariant for $H$, denoted $m_{H}$, and prove that the value of this invariant is connected to the order of $S$. In the case where $\operatorname{char}k=0$, it is shown that if $S$ has finite order then it is either the identity or has order $2m_{H}$. If in addition $H$ is assumed to be coradically graded, it is shown that the order of $S$ is finite if and only if $m_{H}$ is finite. We also consider the case where $\operatorname{char}k=p>0$, generalising the results of $[7]$ to the infinite-dimensional setting.