Weak Hopf algebras and some new solutions of the quantum Yang-Baxter equation
type: Item
instance of: scholarly article
title: Weak Hopf algebras and some new solutions of the quantum Yang-Baxter equation
zbMATH Open document ID: 0916.16020
author: Fang Li
published in: Journal of Algebra
publication date: 1999-07-12T00:00:00Z
review text: A bialgebra \(H\) is called a weak Hopf algebra if there is a linear map \(T\) from \(H\) to \(H\) such that \(I*T*I=I\) and \(T*I*T=T\), \(*\) the convolution product. This generalizes the definition of a left Hopf algebra as studied by \textit{J. A. Green, W. D. Nichols} and the reviewer [J. Algebra 65, 399-411 (1980; Zbl 0439.16008)]. The author also defines a quasi-braided almost bialgebra. For a weak Hopf algebra, he constructs a quantum quasi-double which is a quasi-braided almost bialgebra. He uses this to show for some weak Hopf algebras, one can find some new solutions of the quantum Yang-Baxter equation, in particular when the weak Hopf algebra is a finite Clifford monoid algebra. A semigroup is called Clifford if it is regular and all its idempotents lie in its center. Unfortunately, no examples of finite Clifford monoids are given, nor any examples of weak Hopf algebras, which might lead to new solutions of the quantum Yang-Baxter equation.
reviewed by: Earl J. Taft
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 20M25
zbMATH DE Number: 1221656
zbMATH Keywords: quantum quasi-doubles
zbMATH Keywords: quasi-braided almost bialgebras
zbMATH Keywords: weak Hopf algebras
zbMATH Keywords: quantum Yang-Baxter equation
zbMATH Keywords: finite Clifford monoid algebras
MaRDI profile type: Publication
full work available at URL: jabr.1998.7491
OpenAlex ID: W2057649423
cites work: Q3933122
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cites work: Left Hopf algebras
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cites work: Biinvertible actions of Hopf algebras
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DOI: 10.1006/JABR.1998.7491
description: scientific article; zbMATH DE number 1221656