Generalized Drinfel'd-Sokolov hierarchies
type: Item
instance of: scholarly article
title: Generalized Drinfel'd-Sokolov hierarchies
zbMATH Open document ID: 0749.35044
DOI: 10.1007/BF02099281
author: Mark F. de Groot
author: Timothy J. Hollowood
author: J. Luis Miramontes
published in: Communications in Mathematical Physics
publication date: 1992-09-26T00:00:00Z
review text: During the recent 25 years, the theory of nonlinear partial differential equations of evolution type has undergone a vast development. The crucial discovery that many of the well-known equations in mathematical physics (e.g., the KdV equation, the nonlinear Schrödinger equation, the sine- Gordon equation, and others) are exactly solvable, and can be interpreted in the framework of completely integrable Hamiltonian systems, has led to wide-spread generalizations. In particular, the possibility of exhibiting many of those evolution equations as members of a whole hierarchy defined by equations for matrices of (pseudo-) differential operators of Lax type revealed an evidently deep relationship to algebraic geometry, on the one hand, and to the theory of infinite-dimensional Lie algebras, on the other hand. The approach of constructing integrable hierarchies of evolution equations defined by matrix Lax equations for certain Lie algebras has been studied, in recent years, from various aspects and by several authors, especially with a view towards their applications to conformal quantum field theory and quantum gravity. One of the main contributions, in this regard, was the generalization of the Korteweg-de Vries hierarchy to certain classes of simple Lie algebras by \textit{V. G. Drinfel'd} and \textit{V. V. Sokolov} [Sov. Math. Dokl. 23, 457-462 (1981), transl. from Dokl. Akad. Nauk. SSSR 258, 11-16 (1981; Zbl 0513.35073)], as well as the established association between derived KdV hierarchies and untwisted affine Lie algebras by \textit{G. Wilson} [Ergodic Theory Dyn. Syst. 1, 361-380 (1981; Zbl 0495.58008)]. In the present paper, the authors describe a method of further generalizing the construction of integrable hierarchies of KdV-like equations to (rather) arbitrary untwisted Kac-Moody algebras and their maximal Heisenberg subalgebras. Their main result is that for each positive element of the Heisenberg subalgebra of an untwisted Kac-Moody algebra, there can be ass…
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 17B67
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 37K10
Mathematics Subject Classification ID: 17B81
zbMATH DE Number: 54768
zbMATH Keywords: construction of integrable hierarchies
zbMATH Keywords: soliton equations
zbMATH Keywords: Hamiltonian systems
zbMATH Keywords: Lax equations
zbMATH Keywords: KdV-like equations
zbMATH Keywords: untwisted Kac-Moody algebras
zbMATH Keywords: conjugacy classes of the Weyl group
zbMATH Keywords: Heisenberg subalgebras
zbMATH Keywords: Weyl group
reviewed by: Werner Kleinert
MaRDI profile type: Publication
cites work: Q3679549
cites work: Lie algebras and equations of Korteweg-de Vries type
cites work: The modified Lax and two-dimensional Toda lattice equations associated with simple Lie algebras
cites work: Strings in less than one dimension and the generalized KdV hierarchies
cites work: Q3693672
cites work: Automorphisms of finite order of semisimple Lie algebras
cites work: Q5668957
cites work: A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. I
cites work: Derivative nonlinear Schrodinger equations and Hermitian symmetric spaces
cites work: Q3032461
cites work: Infinite-dimensional Lie groups and algebraic geometry in soliton theory
full work available at URL: https://doi.org/10.1007/bf02099281
OpenAlex ID: W2012157975
description: scientific article; zbMATH DE number 54768