The universal \(R\)-matrix for quantum untwisted affine Lie algebras
type: Item
instance of: scholarly article
title: The universal \(R\)-matrix for quantum untwisted affine Lie algebras
zbMATH Open document ID: 0758.17011
DOI: 10.1007/BF01077085
publication date: 1993-01-16T00:00:00Z
review text: Let \({\mathfrak g}\) be a simple finite dimensional Lie algebra over the complex field. It is well-known that the standard quantum deformation of the universal enveloping algebra \(U_ q({\mathfrak g})\) is a quasitriangular Hopf algebra, i.e. there exists the universal \(R\)-matrix satisfying certain conditions, namely \(\Delta'(x)R=R\Delta(x)\), \(x\in U_ q({\mathfrak g})\), \((\Delta\otimes id)R=R^{13}R^{23}\), \((id\otimes\Delta)R=R^{13}R^{12}\), where \(\Delta\) is the comultiplication in \(V_ q({\mathfrak g})\), \(\Delta'\) is the opposite comultiplication. The explicit formula for \(R\) in the finite dimensional case was known (see references in the paper). The authors deal with an infinite dimensional case. They consider affine untwisted Lie algebras with the symmetrizable Cartan matrix. Let \(\hat{\mathfrak g}\) be such an algebra. Using the so-called normal ordering on the set of positive roots of \(\hat {\mathfrak g}\), one can introduce Cartan-Weil generators in \(U_ q(\hat g)\) and define commutation relations between them. This enables the authors to compute the universal \(R\)-matrix for \(U_ q(\hat{\mathfrak g})\) explicitly.
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 16T20
zbMATH DE Number: 95237
zbMATH Keywords: universal \(R\)-matrix
zbMATH Keywords: quasitriangular Hopf algebra
zbMATH Keywords: affine untwisted Lie algebras
zbMATH Keywords: normal ordering
author: Valeriy N. Tolstoy
author: Sergey Khoroshkin
reviewed by: A. Stolin
MaRDI profile type: Publication
cites work: A q-difference analogue of \(U({\mathfrak g})\) and the Yang-Baxter equation
cites work: Quantum R matrix for the generalized Toda system
cites work: An analogue of P.B.W. theorem and the universal R-matrix for \(U_ h\mathfrak{sl}(N+1)\)
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 95237