1st Edition

Lie Algebraic Methods in Integrable Systems

By Amit K. Roy-Chowdhury Copyright 1999
    366 Pages
    by Chapman & Hall

    368 Pages
    by Chapman & Hall

    Over the last thirty years, the subject of nonlinear integrable systems has grown into a full-fledged research topic. In the last decade, Lie algebraic methods have grown in importance to various fields of theoretical research and worked to establish close relations between apparently unrelated systems.

    The various ideas associated with Lie algebra and Lie groups can be used to form a particularly elegant approach to the properties of nonlinear systems. In this volume, the author exposes the basic techniques of using Lie algebraic concepts to explore the domain of nonlinear integrable systems. His emphasis is not on developing a rigorous mathematical basis, but on using Lie algebraic methods as an effective tool.

    The book begins by establishing a practical basis in Lie algebra, including discussions of structure Lie, loop, and Virasor groups, quantum tori and Kac-Moody algebras, and gradation. It then offers a detailed discussion of prolongation structure and its representation theory, the orbit approach-for both finite and infinite dimension Lie algebra. The author also presents the modern approach to symmetries of integrable systems, including important new ideas in symmetry analysis, such as gauge transformations, and the "soldering" approach. He then moves to Hamiltonian structure, where he presents the Drinfeld-Sokolov approach, the Lie algebraic approach, Kupershmidt's approach, Hamiltonian reductions and the Gelfand Dikii formula. He concludes his treatment of Lie algebraic methods with a discussion of the classical r-matrix, its use, and its relations to double Lie algebra and the KP equation.

    Lax Equation and IST
    Conserved Densities and Hamiltonian Structure
    Symmetry Aspects
    Structure Constants and Basis of Lie Algebra
    Lie Groups and Lie Algebra
    Representation of a Lie Algebra
    Cartan-Killing Form
    Roots Space Decomposition
    Lie Groups: Finite and Infinite Dimensional
    Loop Groups
    Virasoro Group
    Quantum Tori Algebra
    Kac-Moody Algebra
    Serre's Approach to Kac-Moody Algebra
    Other Infinite Dimensional Lie Algebras
    Sectioning of Forms
    The KdV Problem
    The Method of the Hall Structure
    Prolongation in (2+1) Dimension
    Method of Pseudopotentials
    Prolongation Structure and the Bäcklund Transformation
    Constant Coefficient Ideal
    Morphisms and Prolongation
    Principal Prolongation Structure
    Prolongations and Isovectors
    Vessiot's Approach
    The Kac-Moody Algebra
    Integrability Theorem: Adler, Kostant, Symes
    Superintegrable Systems
    Nonlinear Partial Differential Equation
    Extended AKS Theorem
    Space-Dependent Integrable Equation
    The Moment Map
    Moment Map in Relation to Integrable Nonlinear Equation
    Co-Adjoint Orbit of the Volterra Group
    Lie Point and Lie Bäcklund Symmetry
    Lie Bäcklund Transformation
    Some New Ideas in Symmetry Analysis
    Non-Local Symmetries
    Drinfeld Sokolob Approach
    The Lie Algebraic Approach
    Example of Hamiltonian Structure and Reduction
    Hamiltonian Reduction in (2+1) Dimension
    Hamiltonian Reduction of Drinfeld and Sokolov
    Kupershmidt's Approach
    Gelfand Dikii Formula
    Trace Identity and Hamiltonian Structure
    Symmetry and Hamiltonian Structure
    Double Lie Algebra
    Classical r-Matrix
    The Use of r-Matrix
    The r-Matrix and KP Equation


    Amit K. Roy-Chowdhury (University of California, Riverside, USA) (Author)

    "Lie theory and algebraic geometry have played a unifying role in integrable theory since its early rebirth some 30 years ago. They have transformed a mosaic of old examples, due to the masters like Hamilton, Jacobi and Kowalewski, and new examples into general methods and statements. The book under review addresses a number of these topics… contains a variety of interesting topics: some are expained in a user-friendly and elementary way, and others are taken directly from research papers."
    -Pierre Van Moerbeke, in The London Mathematical Society