2014/08/31 by Zhihua Wang, Wang, Zhihua, Libin Li +3 · 2 citations
Mathematics · Physics and Astronomy · #16W30 #19A22 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Nonlinear Waves and Solitons #Quantum Algebra (math.QA) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1409.0225
openalex publication_date 2014/08/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we continue our study of the Green rings of finite dimensional pointed Hopf algebras of rank one initiated in \citeWLZ, 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 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.