Lie Algebraic Methods in Integrable Systems

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 Backlund 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 Backlund Symmetry
Lie Backlund 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