A Lie theoretic Galois theory for the spectral curves of an integrable system. I
type: Item
instance of: scholarly article
title: A Lie theoretic Galois theory for the spectral curves of an integrable system. I
zbMATH Open document ID: 0766.58030
DOI: 10.1007/BF02096626
author: Andrew McDaniel
author: Lawrence Smolinsky
published in: Communications in Mathematical Physics
publication date: 1993-01-16T00:00:00Z
review text: The authors give a classification of the spectral curves for a completely integrable system. They study an integrable system presented in the Lax form \(dA(t,s)/dt=[A(t,s),B(t,s)]\), where \(A(t,s)\), \(B(t,s)\) are elements of a simple complex Lie algebra \(g\) and functions of time \(t\) and rational functions of a point \(s\) of an algebraic curve. For each representation \(\rho\) of \(g\) the spectral curves are defined as irreducible components of a variety presented by an equation \(\text{det}(\rho A(t,s)-z)=0\) which are independent of time. By means of the Galois theory for the spectral curves the authors indicate these curves for generic initial conditions, of the Lax equation by the conjugacy classes of parabolic subgroups of the Weyl group for the appropriate Lie algebra. These results are illustrated by detailed example of the generalized periodic Toda lattice.
reviewed by: E. D. Belokolos
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 37K10
zbMATH DE Number: 95697
zbMATH Keywords: Lie group
zbMATH Keywords: spectral curves
zbMATH Keywords: completely integrable system
zbMATH Keywords: Lie algebra
zbMATH Keywords: periodic Toda lattice
MaRDI profile type: Publication
cites work: Completely integrable systems, Euclidean Lie algebras, and curves
cites work: Linearization of Hamiltonian systems, Jacobi varieties and representation theory
cites work: The Kowalewski top 99 years later: a Lax pair, generalizations and explicit solutions
cites work: Q3222341
cites work: Die Fundamentalgruppe des Raumes der regulären Orbits einer endlichen komplexen Spiegelungsgruppe
cites work: Komplexe Räume
cites work: Linearizing Flows and a Cohomological Interpretation of Lax Equations
cites work: Q5515607
cites work: Q4143433
cites work: Q3488296
cites work: Elliptic solutions of the Kadomtsev-Petviashvili equation and integrable systems of particles
cites work: Representations of sl(n,\({\mathbb{C}})\) and the Toda lattice
cites work: Q3352131
cites work: Theory of nonlinear lattices
cites work: The spectrum of Jacobi matrices
cites work: The spectrum of difference operators and algebraic curves
cites work: Method of the inverse scattering problem with spectral parameter on an algebraic curve
Recommended article: A Lie theoretic Galois theory for the spectral curves of an integrable system. II
Recommended article: Q3739583
Recommended article: Lie algebras and Lax equations with spectral parameter on an elliptic curve
Recommended article: Generalized Jacobians of spectral curves and completely integrable systems
Recommended article: Integrable Systems and Difierential Galois Theory
Recommended article: Algebraic integrable systems related to spectral curves with automorphisms
Recommended article: Q4793220
Recommended article: Bifurcation diagram and the discriminant of a spectral curve of integrable systems on Lie algebras
Recommended article: Differential Galois theory and Darboux transformations for integrable systems
Recommended article: Integrable systems on \(\text{so}(n)\)-valued Lie algebras on higher genus curves
description: scientific article; zbMATH DE number 95697