The quantum Weyl group and the universal quantum \(R\)-matrix for affine Lie algebra \(A^{(1)}_ 1\)
type: Item
instance of: scholarly article
title: The quantum Weyl group and the universal quantum \(R\)-matrix for affine Lie algebra \(A^{(1)}_ 1\)
zbMATH Open document ID: 0776.17011
DOI: 10.1007/BF00777372
author: V. A. Stukopin
author: Sergei Levendorskiĭ
author: Yan Soibelman
published in: Letters in Mathematical Physics
publication date: 1993-12-05T00:00:00Z
review text: As is well-known, one of the main achievements of the theory of quantum groups is to provide large families of solutions of the quantum Yang- Baxter equation, by means of the so-called universal \(R\)-matrix; recall that \(R\) is an element of \(U_ h({\mathfrak g})\hat\otimes U_ h({\mathfrak g})\), \(\mathfrak g\) a finite-dimensional complex Lie algebra. Whereas the existence of \(R\) was from the beginning theoretically established by Drinfel'd, its explicit formula was given first by \textit{V. G. Drinfel'd} himself [Proc. Int. Congr. Math., Berkeley 1986, Vol. 1, 798-820 (1987; Zbl 0667.16003)] for \(s\ell(2)\), then by \textit{M. Rosso} [Commun. Math. Phys. 124, 307-318 (1989; Zbl 0694.17006)] for \(s\ell(N)\) and finally by \textit{S. Levendorskij} and \textit{Ya. S. Soibel'man} [J. Geom. Phys. 7, 241- 254 (1990; Zbl 0729.17009)], and independently by \textit{A. Kirillov} and \textit{N. Yu. Reshetikhin} [Commun. Math. Phys. 134, 421-431 (1991; Zbl 0723.17014)], for the general case. The main tool of the proof in the last two papers is the ``quantum Weyl group'', a Hopf algebra built from both \(U_ h({\mathfrak g})\) and the group algebra of the braid group; the last acts on the first by (a version of) certain algebra automorphisms introduced by \textit{G. Lusztig} [Adv. Math. 70, 237-249 (1988; Zbl 0651.17007)] (the quantum Weyl group was first considered in [\textit{Ya. S. Soibel'man}, Algebra Anal. 2, 190-212 (1990; Zbl 0708.46029)]. However, a proof of the explicit formula for the universal \(R\)-matrix using instead of the quantum Weyl group, a ``quantum'' variant of the Cartan-Weyl basis was offered by \textit{S. M. Khoroshkin} and \textit{V. N. Tolstoj} [Funkts. Anal. Prilozh. 26, 85-88 (1992; Zbl 0758.17011); see also Commun. Math. Phys. 141, 599-617 (1991; Zbl 0744.17015)] who showed also that their approach is also available for \(U_ h({\mathfrak g})\), if \(\mathfrak g\) is an affine non-twisted Kac-Moody algebra. In the paper under review, the expli…
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 17B67
zbMATH DE Number: 202408
zbMATH Keywords: quantum groups
zbMATH Keywords: quantum Yang-Baxter equation
zbMATH Keywords: universal \(R\)-matrix
zbMATH Keywords: affine Kac-Moody algebra
zbMATH Keywords: braid group automorphisms
Wikidata QID: Q115394877
reviewed by: Nicolás Andruskiewitsch
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00777372
OpenAlex ID: W2940964784
description: scientific article; zbMATH DE number 202408