Green rings of pointed rank one Hopf algebras of non-nilpotent type

Abstract

In this paper, we continue our study of the Green rings of finite dimensional pointed Hopf algebras of rank one initiated in [22], 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 G0(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 in the stable category of H. Finally, as an example, we present both the Green ring and the Grothendieck ring of the Radford Hopf algebra in terms of generators and relations.