Presents the basic techniques of using Lie algebraic concepts to explore the domain of nonlinear integrable systems. This 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.

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

"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