Generalized Knizhnik-Zamolodchikov equations and the factorization of their solutions
type: Item
instance of: scholarly article
title: Generalized Knizhnik-Zamolodchikov equations and the factorization of their solutions
zbMATH Open document ID: 0829.35124
DOI: 10.1007/BF01208387
author: Valentina Golubeva
author: V. P. Leksin
published in: Mathematical Notes
publication date: 1995-09-12T00:00:00Z
review text: Generalized Knizhnik-Zamolodchikov equations represent the class of Fuchsian systems on \(\mathbb{C}^n\), \(n \geq 2\), that are invariant under the complexification of the action of the Weyl group for a root system in \(\mathbb{R}^n\). The peculiarities of these equations consist in the complexification of the union of hyperplanes orthogonal to the roots of a given root system. Kohno noted that the coefficients of these equations are classical \(r\)-matrices and proposed to construct such equations by the Belavin-Drinfeld method for the theory of \(r\)-matrices. Cherednik established the connection of classical \(r\)-matrices with systems of roots of type \(A_n\), \(n \geq 2\), introduced the concept of a \(r\)-matrix for an arbitrary system of roots, and also deduced an analog of the Knizhnik-Zamolodchikov equation for an arbitrary system of roots. In the present note we show that, as in the case considered by Kohno, the coefficients of simplest generalized Knizhnik-Zamolodchikov equations are generalized \(r\)-matrices. Moreover, we prove the equivalence of the complete integrability conditions for generalized Knizhnik-Zamolodchikov equations and for infinitesimal relations in generalized pure braid groups. Finally, we specify Cherednik's assertion on the local behavior of fundamental solutions of generalized Knizhnik-Zamolodchikov equations and the monodromy of these solutions.
Mathematics Subject Classification ID: 35Q58
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 37K10
Mathematics Subject Classification ID: 20F36
zbMATH DE Number: 795596
zbMATH Keywords: generalized \(r\)-matrices
zbMATH Keywords: Yang-Baxter equations
zbMATH Keywords: Fuchsian systems
zbMATH Keywords: root system
zbMATH Keywords: complete integrability conditions
zbMATH Keywords: braid groups
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 795596