\(R\)-matrix arising from affine Hecke algebras and its application to Macdonald's difference operators
type: Item
instance of: scholarly article
title: \(R\)-matrix arising from affine Hecke algebras and its application to Macdonald's difference operators
zbMATH Open document ID: 0820.17023
DOI: 10.1007/BF02099422
published in: Communications in Mathematical Physics
publication date: 1994-12-18T00:00:00Z
review text: A construction of \(R\)-matrices in the sense of Cherednik is presented for an arbitrary affine root system. It means that for any root \(\alpha\) an operator \(R_ \alpha\) is constructed acting on a \(| W|\)- dimensional space \(V\) over a certain field of fractions of the group algebra of a lattice \(L\), including the root lattice and being contained in the weight lattice; \(W\) is the Weyl group of the underlying finite- dimensional Lie algebra. The operators \(R_ \alpha\), \(R_ \beta\) satisfy the analog of the Yang-Baxter equation defined by the structure of the corresponding rank two root system. The construction is based on the two actions of a Hecke algebra on \(V\). A \(q\)-analog of the Knizhnik- Zamolodchikov equations is written down by means of the corresponding \(R\). Cherednik's equations are their specializations. It is shown that the solutions are eigenvectors of Macdonald's difference operators.
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 35Q40
Mathematics Subject Classification ID: 17B67
Mathematics Subject Classification ID: 81R50
zbMATH DE Number: 694967
zbMATH Keywords: affine Hecke algebra
zbMATH Keywords: \(R\)-matrices
zbMATH Keywords: affine root system
zbMATH Keywords: Knizhnik- Zamolodchikov equations
zbMATH Keywords: Macdonald's difference operators
author: Shin-Ichi Kato
reviewed by: Sergei Khoroshkin
MaRDI profile type: Publication
cites work: Q5580319
cites work: Factorizing particles on a half-line and root systems
cites work: Quantum Knizhnik-Zamolodchikov equations and affine root systems
cites work: Q4024104
cites work: Quantum affine algebras and holonomic difference equations
cites work: A \(q\)-analogue of \(U(\mathfrak{gl}(N+1))\), Hecke algebra, and the Yang-Baxter equation
cites work: On the Kazhdan-Lusztig polynomials for affine Weyl groups
cites work: Q3964803
cites work: Affine Hecke Algebras and Their Graded Version
cites work: Affine root systems and Dedekind's \(\eta\)-function
cites work: The Poincaré series of a Coxeter group
cites work: Analyse harmonique dans les systèmes de Tits bornologiques de type affine
cites work: Integrable connections related to zonal spherical functions
full work available at URL: https://doi.org/10.1007/bf02099422
OpenAlex ID: W2076291133
description: scientific article; zbMATH DE number 694967