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.

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.

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