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Please use this identifier to cite or link to this item: http://hdl.handle.net/1942/20229

Title: Green rings of pointed rank one Hopf algebras of non-nilpotent type
Authors: Wang, Zhihua
Li, Libin
Zhang, Yinhuo
Issue Date: 2016
Citation: JOURNAL OF ALGEBRA, 449, p. 108-137
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.
Notes: Wang, ZH (reprint author), Nanjing Normal Univ, Taizhou Coll, Dept Math, Taizhou 225300, Peoples R China. mafzhua@126.com; lbli@yzu.edu.cn; yinhuo.zang@hassolt.bo
URI: http://hdl.handle.net/1942/20229
DOI: 10.1016/j.jalgebra.2015.11.002
ISI #: 000375634600004
ISSN: 0021-8693
Category: A1
Type: Journal Contribution
Validation: ecoom, 2017
Appears in Collections: Research publications

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