Gauss decomposition of connection matrices and application to Yang-Baxter equation. II
type: Item
instance of: scholarly article
title: Gauss decomposition of connection matrices and application to Yang-Baxter equation. II
zbMATH Open document ID: 0819.33010
author: Kazuhiko Aomoto
author: Yoshifumi Kato
published in: Proceedings of the Japan Academy. Series A
publication date: 1995-08-08T00:00:00Z
review text: [For part I see the authors' paper in ibid. 69, No. 7, 238-242 (1993; Zbl 0797.33011).] It is the second part of the paper consisting of two parts. Here the authors study matrix functions introduced in the first part. Basic properties of the principal connection matrix \(G= G(x\mid \alpha_ 1)\), \(x= (x_ 1, x_ 2)\), are given. This matrix function depends only on \(x_ 2/ x_ 1\). The formula is presented relating \(G= G(x\mid \alpha_ 1)\) with \(G= G(x^{-1} \mid \alpha_ 1^ \prime)\). The special cycles \(\text{reg } Y^ -_ i \otimes Y^ +_{ n-i}\) are defined which are connected with \(G\). It is given the asymptotics characterizing them. The cycle \(\text{reg } Y^ -_ i \otimes Y^ +_{n -i}\) is expressed as a linear combination of \(\text{reg } Y^ -_{j, n-j}\) and \(\text{reg } Y^ +_{n-i, n-j}\). Explicit expressions for the coefficients are given in terms of theta monomials. Matrix elements of the matrix \(W_{\tau_ r}\), introduced in the first part of the paper, is expressed in terms of the connection matrix for Jackson integrals of the function depending only on \(x_ r\) and \(x_{r+1}\). The authors obtain the one cocycle condition for \(\{ W^{(m) }_{\tau_ r} \}_{r=1}^{m -1}\) which coincides with the Yang-Baxter equation.
Mathematics Subject Classification ID: 33D10
zbMATH DE Number: 549047
zbMATH Keywords: Jackson integrals
zbMATH Keywords: Yang-Baxter equation
reviewed by: A. U. Klimyk
MaRDI profile type: Publication
full work available at URL: pjaa.69.341
OpenAlex ID: W4256164502
cites work: Gauss decomposition of connection matrices and application to Yang-Baxter equation. I
cites work: Quantum Knizhnik-Zamolodchikov equations and affine root systems
cites work: Quantum Knizhnik-Zamolodchikov equations and holomorphic vector bundles
cites work: Quantum affine algebras and holonomic difference equations
cites work: Quantum algebra structure of certain Jackson integrals
cites work: Holonomic \(q\)-difference system of the first order associated with a Jackson integral of Selberg type
cites work: Quantized Knizhnik-Zamolodchikov equations, quantum Yang-Baxter equation, and difference equations for \(q\)-hypergeometric functions
DOI: 10.3792/PJAA.69.341
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description: scientific article; zbMATH DE number 549047