Lie Algebraic Methods in Integrable Systems

ISBN-13: 9781584880370 / Angielski / Miękka / 1999 / 368 str.

Roy Chowdhury

Opis

The various ideas associated with Lie algebra and Lie groups can be us ed to form a particularly elegant approach to the properties of nonlin ear systems. In this volume, the author exposes the basic techniques o f using Lie algebraic concepts to explore the domain of nonlinear inte grable systems. His emphasis is not on developing a rigorous mathemati cal basis, but on using Lie algebraic methods as an effective tool.

Recenzje

"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

Spis treści

INTRODUCTION
Lax Equation and IST
Conserved Densities and Hamiltonian Structure
Symmetry Aspects
Observations
LIE ALGEBRA
Introduction
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
Gradation
Other Infinite Dimensional Lie Algebras
PROLONGATION THEORY
Introduction
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
Connections
Morphisms and Prolongation
Principal Prolongation Structure
Prolongations and Isovectors
Vessiot's Approach
Observations
CO-ADJOINT ORBITS
Introduction
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
SYMMETRIES OF INTEGRABLE SYSTEMS
Introduction
Lie Point and Lie Bäcklund Symmetry
Lie Bäcklund Transformation
Some New Ideas in Symmetry Analysis
Non-Local Symmetries
Observations
HAMILTONIAN STRUCTURE
Introduction
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
CLASSICAL r-MATRIX
Introduction
Double Lie Algebra
Classical r-Matrix
The Use of r-Matrix
The r-Matrix and KP Equation

O autorze

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