On the Antipode of Hopf Algebras with the Dual Chevalley Property

TL;DR

This paper investigates the antipode of finite-dimensional Hopf algebras with the dual Chevalley property, deriving an annihilation polynomial and characterizations of the quasi-exponent, revealing structural properties related to the coradical and Loewy length.

Contribution

It introduces an annihilation polynomial for the antipode of such Hopf algebras and characterizes the quasi-exponent, linking algebraic properties to the coradical exponent and Loewy length.

Findings

Order of S^2 divides N in characteristic 0

Annihilation polynomial determined by coradical exponent and Loewy length

Two characterizations of the quasi-exponent

Abstract

In this paper, we study the antipode of a finite-dimensional Hopf algebra $H$ with the dual Chevalley property and obtain an annihilation polynomial for its antipode $S$. The annihilation polynomial is determined by the exponent $N$ of the coradical and the Loewy length. In particular, the order of $S^2$ divides $N$ in characteristic $0$. Moreover, we get two characterizations of the quasi-exponent.