Special biserial coalgebras and representations of quantum \(SL(2)\).
type: Item
instance of: scholarly article
title: Special biserial coalgebras and representations of quantum \(SL(2)\).
zbMATH Open document ID: 1262.16034
author: William Chin
published in: Journal of Algebra
publication date: 2012-08-23T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0609370
review text: Let \(K\) be a field. Following the definitions of a special biserial finite-dimensional \(K\)-algebra and a string \(K\)-algebra, the author introduces and studies the class of special biserial \(K\)-coalgebras and string \(K\)-coalgebras. Relations between these concepts and the concept of a string coalgebra introduced by the reviewer [in Lect. Notes Pure Appl. Math. 236, 465-492 (2004; Zbl 1072.16015)] are discussed, elementary properties of special biserial \(K\)-coalgebras are proved, and the theory of special biserial \(K\)-coalgebras is developed in the paper. The results are used to get an explicit description of finite-dimensional comodules and the Auslander-Reiten quiver for the coordinate Hopf algebra \(C\) of the quantum \(SL(2)\) at a root of unity. In particular, it is shown that \(C\), viewed as coalgebra, is tame of discrete comodule type. The stable Green ring of \(C\) and the quantum dimensions of indecomposable \(C\)-comodules are computed. In particular, it is shown that every finite-dimensional non-injective indecomposable \(C\)-comodule is of non-zero quantum dimension.
reviewed by: Daniel Simson
Mathematics Subject Classification ID: 16T15
Mathematics Subject Classification ID: 16T20
Mathematics Subject Classification ID: 16G60
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 05C20
Mathematics Subject Classification ID: 16G20
zbMATH DE Number: 6070504
zbMATH Keywords: biserial coalgebras
zbMATH Keywords: string coalgebras
zbMATH Keywords: quantum groups
zbMATH Keywords: Hopf algebras
zbMATH Keywords: quiver algebras
zbMATH Keywords: stable Green rings
zbMATH Keywords: finite-dimensional comodules
zbMATH Keywords: Auslander-Reiten quivers
zbMATH Keywords: tame discrete comodule type
zbMATH Keywords: quantum dimensions
MaRDI profile type: Publication
OpenAlex ID: W1987712548
arXiv ID: math/0609370
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DOI: 10.1016/J.JALGEBRA.2011.07.025
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description: scientific article; zbMATH DE number 6070504