Exchange algebra and the Drinfel'd-Sokolov theorem
type: Item
instance of: scholarly article
title: Exchange algebra and the Drinfel'd-Sokolov theorem
zbMATH Open document ID: 0746.17016
DOI: 10.1007/BF02099292
author: S. Singh
published in: Communications in Mathematical Physics
publication date: 1992-06-27T00:00:00Z
review text: The relationship between the exchange algebra (Gervais, Neveu; Babelon), encoding the conformal structure of classical Toda field theory, and the separation of the two chiralities in this integrable field model is analyzed. The chiral halves are described as the solutions of the Drinfel'd-Sokolov linear equations [\textit{V. G. Drinfel'd} and \textit{V. V. Sokolov}, Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 24, 81-180 (1984; Zbl 0558.58027); see also \textit{A. N. Leznov} and \textit{M. V. Saveliev}, Lett. Math. Phys. 3, 489-494 (1979; Zbl 0415.35017)]. The sets of relations (Poisson brackets) defining the normalized chiral exchange algebras are formulated by taking into account the matrix solutions \((r^ +\) and \(r^ -)\) of the Yang-Baxter equations in classical Toda field theory [\textit{J. Balog}, \textit{L. Dabrowski} and \textit{L. Feher}, Phys. Lett., B. 244, 227-234 (1991; Zbl 0724.58069)]. In the framework of the developed construction, by enlarging the phase space and considering the zero modes of the left and right movers as well as their conjugate variables as independent dynamical variables, it is shown that one can introduce a conformal invariant Bloch wave basis such that the two chiralities commute. As a result, the periodic and local solution of Toda field theory is reconstructed in terms of this basis.
Mathematics Subject Classification ID: 17B81
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 37K10
Mathematics Subject Classification ID: 81T40
zbMATH DE Number: 25865
zbMATH Keywords: exchange algebra
zbMATH Keywords: Toda field theory
zbMATH Keywords: Drinfel'd-Sokolov linear equations
zbMATH Keywords: chiral exchange algebras
zbMATH Keywords: Yang-Baxter equations
zbMATH Keywords: Bloch wave basis
Wikidata QID: Q64038605
reviewed by: A. A. Bogush
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 25865