A \(q\)-analogue of \(U(\mathfrak{gl}(N+1))\), Hecke algebra, and the Yang-Baxter equation
type: Item
instance of: scholarly article
title: A \(q\)-analogue of \(U(\mathfrak{gl}(N+1))\), Hecke algebra, and the Yang-Baxter equation
zbMATH Open document ID: 0602.17005
DOI: 10.1007/BF00400222
published in: Letters in Mathematical Physics
publication date: 1986-01-01T00:00:00Z
review text: The following problem about the Yang-Baxter equation (YBE) is studied: given a classical r-matrix (= solution to the classical YBE), is it possible to find the corresponding quantum R-matrix (= solution to the quantum YBE) having the former as its classical limit ? Note that now the classical r-matrices associated with a simple Lie algebra \(\mathfrak g\) are classified. It has been realized that to construct R-matrices one has to ``quantize'' the Lie algebra structure itself. This was worked out by Drinfel'd for a class of rational solutions that generalize Yang's one. Here a \(q\)-analogue \(\hat U(\mathfrak g)\) of the universal enveloping algebra \(U(\mathfrak g)\) appears. In this letter, the structure and the representations of \(\hat U(\mathfrak g)\) in the case \(\mathfrak g=\mathfrak{gl}(N+1)\) are studied having in mind its applications to the YBE.
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 16T25
Mathematics Subject Classification ID: 81R50
zbMATH DE Number: 3970994
zbMATH Keywords: Yang-Baxter equation
zbMATH Keywords: r-matrix
zbMATH Keywords: quantum R-matrix
zbMATH Keywords: universal enveloping algebra
author: M. Jimbo
reviewed by: L. N. Vaserstein
MaRDI profile type: Publication
cites work: On the classical Young-Baxter equation for simple Lie algebras
cites work: Some algebraic structures connected with the Yang-Baxter equation
cites work: Some Exact Results for the Many-Body Problem in one Dimension with Repulsive Delta-Function Interaction
cites work: A q-difference analogue of \(U({\mathfrak g})\) and the Yang-Baxter equation
cites work: Yang-Baxter equation and representation theory. I
cites work: A q-analogue of Young symmetrizer
cites work: Q3324796
description: scientific article; zbMATH DE number 3970994