Green Rings of Pointed Rank One Hopf algebras of Non-nilpotent Type

TL;DR

This paper analyzes the structure of Green rings and Grothendieck rings of finite dimensional pointed rank one Hopf algebras of non-nilpotent type, revealing their algebraic properties and relations.

Contribution

It determines indecomposable modules, describes Green and Grothendieck rings, and explores their algebraic structure and relations for non-nilpotent rank one Hopf algebras.

Findings

Jacobson radical equals the kernel of the Cartan map

Green ring has no non-trivial idempotents

Stable Green ring is a transitive fusion ring

Abstract

In this paper, we continue our study of the Green rings of finite dimensional pointed Hopf algebras of rank one initiated in \cite{WLZ}, but focus on those Hopf algebras of non-nilpotent type. Let $H$ be a finite dimensional pointed rank one Hopf algebra of non-nilpotent type. We first determine all non-isomorphic indecomposable $H$-modules and describe the Clebsch-Gordan formulas for them. We then study the structures of both the Green ring $r(H)$ and the Grothendieck ring $G_0(H)$ of $H$ and establish the precise relation between the two rings. We use the Cartan map of $H$ to study the Jacobson radical and the idempotents of $r(H)$. It turns out that the Jacobson radical of $r(H)$ is exactly the kernel of the Cartan map, a principal ideal of $r(H)$, and $r(H)$ has no non-trivial idempotents. Besides, we show that the stable Green ring of $H$ is a transitive fusion ring. This enables us to calculate Frobenius-Perron dimensions of objects of the stable category of $H$. Finally, as an example, we present both the Green ring and the Grothendieck ring of the Radford Hopf algebra.