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Green Rings of Pointed Rank One Hopf algebras of Nilpotent Type

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Abstract

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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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Hasselt, 3590, Diepeenbeek, Belgium

    Zhihua Wang & Yinhuo Zhang

  2. School of Mathematical Science, Yangzhou University, Yangzhou, 225002, China

    Zhihua Wang & Libin Li

Authors
  1. Zhihua Wang
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  2. Libin Li
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  3. Yinhuo Zhang
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Correspondence to Yinhuo Zhang.

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Presented by Susan Montgomery

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Wang, Z., Li, L. & Zhang, Y. Green Rings of Pointed Rank One Hopf algebras of Nilpotent Type. Algebr Represent Theor 17, 1901–1924 (2014). https://doi.org/10.1007/s10468-014-9484-9

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  • DOI: https://doi.org/10.1007/s10468-014-9484-9

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