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

摘要

In this paper, we continue our study of the Green rings of finite dimensionalpointed Hopf algebras of rank one initiated in \cite{WLZ}, but focus on thoseHopf algebras of non-nilpotent type. Let $H$ be a finite dimensional pointedrank one Hopf algebra of non-nilpotent type. We first determine allnon-isomorphic indecomposable $H$-modules and describe the Clebsch-Gordanformulas 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 relationbetween the two rings. We use the Cartan map of $H$ to study the Jacobsonradical and the idempotents of $r(H)$. It turns out that the Jacobson radicalof $r(H)$ 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 stableGreen ring of $H$ is a transitive fusion ring. This enables us to calculateFrobenius-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 theRadford Hopf algebra.