Contents. Preface...VII. Introduction... 1

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1 Preface...VII Introduction... 1 I Preliminaries LieGroupsandLieAlgebras Lie Groups and an Infinite-Dimensional Setting TheLieAlgebraofaLieGroup The Exponential Map AbstractLieAlgebras Adjoint and Coadjoint Orbits The Adjoint Representation The Coadjoint Representation CentralExtensions LieAlgebraCentralExtensions CentralExtensionsofLieGroups The Euler Equations for Lie Groups PoissonStructuresonManifolds Hamiltonian Equations on the Dual of a Lie Algebra A Riemannian Approach to the Euler Equations Poisson Pairs and Bi-Hamiltonian Structures Integrable Systems and the Liouville Arnold Theorem Symplectic Reduction Hamiltonian Group Actions Symplectic Quotients Bibliographical Notes II Infinite-Dimensional Lie Groups: Their Geometry, Orbits, and Dynamical Systems LoopGroupsandAffineLieAlgebras The Central Extension of the Loop Lie algebra

2 X 1.2 Coadjoint Orbits of Affine Lie Groups Construction of the Central Extension of the Loop Group Bibliographical Notes Diffeomorphisms of the Circle and the Virasoro Bott Group CentralExtensions Coadjoint Orbits of the Group of Circle Diffeomorphisms The Virasoro Coadjoint Action and Hill s Operators The Virasoro Bott Group and the Korteweg de Vries Equation The Bi-Hamiltonian Structure of the KdV Equation Bibliographical Notes Groups of Diffeomorphisms The Group of Volume-Preserving Diffeomorphisms and Its Coadjoint Representation The Euler Equation of an Ideal Incompressible Fluid The Hamiltonian Structure and First Integrals of the Euler Equations for an Incompressible Fluid Semidirect Products: The Group Setting for an Ideal Magnetohydrodynamics and Compressible Fluids Symplectic Structure on the Space of Knots and the Landau Lifschitz Equation Diffeomorphism Groups as Metric Spaces Bibliographical Notes The Group of Pseudodifferential Symbols The Lie Algebra of Pseudodifferential Symbols Outer Derivations and Central Extensions of ψ DS The Manin Triple of Pseudodifferential Symbols The Lie Group of α-pseudodifferential Symbols The Exponential Map for Pseudodifferential Symbols Poisson Structures on the Group of α-pseudodifferential Symbols Integrable Hierarchies on the Poisson Lie Group G INT Bibliographical Notes Double Loop and Elliptic Lie Groups Central Extensions of Double Loop Groups andtheirliealgebras Coadjoint Orbits Holomorphic Loop Groups and Monodromy Digression: Definition of the Calogero Moser Systems The Trigonometric Calogero Moser System andaffineliealgebras The Elliptic Calogero Moser System and Elliptic Lie Algebras Bibliographical Notes

3 XI III Applications of Groups: Topological and Holomorphic Gauge Theories Holomorphic Bundles and Hitchin Systems Basics on Holomorphic Bundles HitchinSystems Bibliographical Notes PoissonStructuresonModuliSpaces Moduli Spaces of Flat Connections on Riemann Surfaces Poincaré Residue and the Cauchy Stokes Formula Moduli Spaces of Holomorphic Bundles Bibliographical Notes Around the Chern Simons Functional A Reminder on the Lagrangian Formalism The Topological Chern Simons Action Functional The Holomorphic Chern Simons Action Functional A Reminder on Linking Numbers The Abelian Chern Simons Path Integral and Linking Numbers Bibliographical Notes Polar Homology Introduction to Polar Homology Polar Homology of Projective Varieties Polar Intersections and Linkings Polar Homology for Affine Curves Bibliographical Notes Appendices A.1 RootSystems FiniteRootSystems Semisimple Complex Lie Algebras Affine and Elliptic Root Systems Root Systems and Calogero Moser Hamiltonians A.2 Compact Lie Groups The Structure of Compact Groups A Cohomology Generator for a Simple Compact Group 224 A.3 Krichever NovikovAlgebras Holomorphic Vector Fields on C and the Virasoro Algebra Definition of the Krichever Novikov Algebras and Almost Grading CentralExtensions Affine Krichever Novikov Algebras, Coadjoint Orbits, and Holomorphic Bundles A.4 Kähler Structures on the Virasoro and Loop Group Coadjoint Orbits...234

4 XII 4.1 The Kähler Geometry of the Homogeneous Space Diff(S 1 )/S The Action of Diff(S 1 ) and Kähler Geometry onthebasedloopspaces A.5 Diffeomorphism Groups and Optimal Mass Transport The Inviscid Burgers Equation as a Geodesic Equation on the Diffeomorphism Group Metric on the Space of Densities and the Otto Calculus The Hamiltonian Framework of the Riemannian Submersion A.6 Metrics and Diameters of the Group of Hamiltonian Diffeomorphisms The Hofer Metric and Bi-invariant Pseudometrics on the Group of Hamiltonian Diffeomorphisms The Infinite L 2 -Diameter of the Group of Hamiltonian Diffeomorphisms A.7 Semidirect Extensions of the Diffeomorphism Group and Gas Dynamics A.8 The Drinfeld Sokolov Reduction The Drinfeld Sokolov Construction The Kupershmidt Wilson Theorem and the Proofs A.9 The Lie Algebra gl The Lie Algebra gl and Its Subalgebras The Central Extension of gl q-difference Operators and gl A.10 Torus Actions on the Moduli Space of Flat Connections Commuting Functions on the Moduli Space The Case of SU(2) SL(n, C) and the Rational Ruijsenaars Schneider System References Index...301

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