Green Rings of Pointed Rank One Hopf Algebras of Nilpotent Type

Zhihua Wang; Libin Li; Yinhuo ZhangSusan Montgomery

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Green Rings of Pointed Rank One Hopf Algebras of Nilpotent Type

机译：Green Rings of Pointed Rank One Hopf Algebras of Nilpotent Type

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摘要

Let H be a finite dimensional pointed rank one Hopf algebra of nilpotent type.We first determine all finite dimensional indecomposable H-modules up to isomorphism, and then establish the Clebsch-Gordan formulas for the decompositions of the tensor products of indecomposable H-modules by virtue of almost split sequences. The Green ring r(H) of H will be presented in terms of generators and relations. It turns out that the Green ring r(H) is commutative and is generated by one variable over the Grothendieck ring G_0(H) of H modulo one relation. Moreover, r(H) is Frobenius and symmetric with dual bases associated to almost split sequences, and its Jacobson radical is a principal ideal. Finally, we show that the stable Green ring, the Green ring of the stable module category, is isomorphic to the quotient ring of r(H) modulo all projective modules. It turns out that the complexified stable Green algebra is a group-like algebra and hence a bi-Frobenius algebra.

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来源
《Algebras and representation theory》 |2014年第6期|1901-1924|共24页
作者
Zhihua Wang; Libin Li; Yinhuo ZhangSusan Montgomery;
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作者单位

Department of Mathematics and Statistics, University of Hasselt, 3590 Diepeenbeek, Belgium;

School of Mathematical Science, Yangzhou University, Yangzhou 225002, China;

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正文语种英语
中图分类抽象代数（近世代数）;
关键词
Green ring; Indecomposable module; Symmetric ring; Jacobson radical; Group-like algebra; bi-Frobenius algebra;

Green Rings of Pointed Rank One Hopf Algebras of Nilpotent Type

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