The Green rings of the generalized Taft Hopf algebras

2012/10/16 by Libin Li, Li, Libin, Yinhuo Zhang +1 · 2 citations
Mathematics · #16W30 #19A22 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Finite Group Theory Research #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1210.4245

openalex publication_date 2012/10/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we investigate the Green ring r(H_n,d) of the generalized Taft algebra H_n,d, extending the results of Chen, Van Oystaeyen and Zhang in \citeCoz. We shall determine all nilpotent elements of the Green ring r(H_n,d). It turns out that each nilpotent element in r(H_n,d) can be written as a sum of indecomposable projective representations. The Jacobson radical J(r(H_n,d)) of r(H_n,d) is generated by one element, and its rank is n-n/d. Moreover, we will present all the finite dimensional indecomposable representations over the complexified Green ring R(H_n,d) of H_n,d. Our analysis is based on the decomposition of the tensor product of indecomposable representations and the observation of the solutions for the system of equations associated to the generating relations of the Green ring r(H_n,d).

Cited by

• The Projective Class Rings of Drinfeld doubles of pointed rank one Hopf algebras
• The Green ring of a restricted enveloping algebra in characteristic 2

Related

• The Green Rings of Taft algebras
• Green rings of Drinfeld Doubles of Taft algebras
• Green Rings of Pointed Rank One Hopf algebras of Non-nilpotent Type
• Green Rings of Finite Dimensional Pointed Rank One Hopf algebras of Nilpotent Type
• Stable Green ring of the Drinfeld doubles of the generalised Taft algebras (corrections and new results)