Lie Algebraic Methods in Integrable Systems

ISBN-13: 9781138401921 / Angielski / Twarda / 2017

Amit K. Roy-Chowdhury

Opis

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.

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 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

O autorze

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