On the set-theoretical Yang-Baxter equation
type: Item
instance of: scholarly article
title: On the set-theoretical Yang-Baxter equation
zbMATH Open document ID: 0960.16043
DOI: 10.1215/S0012-7094-00-10411-5
author: Jiang-Hua Lu
author: Min Yan
author: Yongchang Zhu
published in: Duke Mathematical Journal
publication date: 2001-02-07T00:00:00Z
review text: Let \(G\) be a group with a left and a right action on itself. The left (respectively right) action of \(u\in G\) on \(v\in G\) is denoted by \(u\to v\) (respectively \(v\gets u\)). The main result tells that if the compatibility condition \(uv=(u\to v)(u\gets v)\) holds for any \(u,v\in G\), then \(R\colon G\times G\to G\times G\), \(R(u,v)=(u\gets v,u\to v)\) is an invertible solution of the set-theoretical quantum Yang-Baxter equation on the set \(G\). Two conceptual reformulations of the above conditions on \(G\) are given by using braiding operators and bijective 1-cocycles. It is shown that this construction generalizes the ones of \textit{A. Weinstein} and \textit{P. Xu} [Commun. Math. Phys. 148, No. 2, 309-343 (1992; Zbl 0849.17015)] and \textit{P. Etingof, T. Schedler} and \textit{A. Soloviev} [Duke. Math. J. 100, No. 2, 169-209 (1999)]. A universal construction for non-degenerate solutions of the set-theoretical quantum Yang-Baxter equation is presented.
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 81R50
zbMATH DE Number: 1501663
zbMATH Keywords: quantum Yang-Baxter equation
zbMATH Keywords: braiding operators
zbMATH Keywords: 1-cocycles
zbMATH Keywords: braid equations
reviewed by: Sorin Dascalescu
MaRDI profile type: Publication
cites work: On some unsolved problems in quantum group theory
cites work: A method of construction of finite-dimensional triangular semisimple Hopf algebras
cites work: Set-theoretical solutions to the quantum Yang-Baxter equation
cites work: Classical solutions of the quantum Yang-Baxter equation
description: scientific article; zbMATH DE number 1501663