Preface ....................................................... VII
Introduction .................................................... 1
I Preliminaries ................................................ 7
1 Lie Groups and Lie Algebras .................................. 7
1.1 Lie Groups and an Infinite-Dimensional Setting .......... 7
1.2 The Lie Algebra of a Lie Group .......................... 9
1.3 The Exponential Map .................................... 12
1.4 Abstract Lie Algebras .................................. 15
2 Adjoint and Coadjoint Orbits ................................ 17
2.1 The Adjoint Representation ............................. 17
2.2 The Coadjoint Representation ........................... 19
3 Central Extensions .......................................... 21
3.1 Lie Algebra Central Extensions ......................... 22
3.2 Central Extensions of Lie Groups ....................... 24
4 The Euler Equations for Lie Groups .......................... 26
4.1 Poisson Structures on Manifolds ........................ 26
4.2 Hamiltonian Equations on the Dual of a Lie Algebra ..... 29
4.3 A Riemannian Approach to the Euler Equations ........... 30
4.4 Poisson Pairs and Bi-Hamiltonian Structures ............ 35
4.5 Integrable Systems and the Liouville-Arnold Theorem .... 38
5 Symplectic Reduction ........................................ 40
5.1 Hamiltonian Group Actions .............................. 41
5.2 Symplectic Quotients ................................... 42
6 Bibliographical Notes ....................................... 44
II Infinite-Dimensional Lie Groups: Their Geometry, Orbits,
and Dynamical Systems ....................................... 47
1 Loop Groups and Affine Lie Algebras ......................... 47
1.1 The Central Extension of the Loop Lie algebra .......... 47
1.2 Coadjoint Orbits of Afflne Lie Groups .................. 52
1.3 Construction of the Central Extension of the Loop
Group .................................................. 58
1.4 Bibliographical Notes .................................. 65
2 Diffeomorphisms of the Circle and the Virasoro Bott Group ... 67
2.1 Central Extensions ..................................... 67
2.2 Coadjoint Orbits of the Group of Circle
Diffeomorphisms ........................................ 70
2.3 The Virasoro Coadjoint Action and Hill's Operators ..... 72
2.4 The Virasoro-Bott Group and the Korteweg-de Vries
Equation ............................................... 80
2.5 The Bi-Hamiltonian Structure of the KdV Equation ....... 82
2.6 Bibliographical Notes .................................. 86
3 Groups of Diffeomorphisms ................................... 88
3.1 The Group of Volume-Preserving Diffeomorphisms
and Its Coadjoint Representation ....................... 88
3.2 The Euler Equation of an Ideal Incompressible Fluid .... 90
3.3 The Hamiltonian Structure and First Integrals
of the Euler Equations for an Incompressible Fluid ..... 91
3.4 Semidirect Products: The Group Setting for an Ideal
Magnetohydrodynamics and Compressible Fluids ........... 95
3.5 Symplectic Structure on the Space of Knots and the
Landau-Lifschitz Equation .............................. 99
3.6 Diffeomorphism Groups as Metric Spaces ................ 105
3.7 Bibliographical Notes ................................. 109
4 The Group of Pseudodifferential Symbols .................... 111
4.1 The Lie Algebra of Pseudodifferential Symbols ......... 111
4.2 Outer Derivations and Central Extensions of ψ DS ...... 113
4.3 The Manin Triple of Pseudodifferential Symbols ........ 117
4.4 The Lie Group of α-Pseudodifferential Symbols ......... 119
4.5 The Exponential Map for Pseudodifferential Symbols .... 122
4.6 Poisson Structures on the Group of
α-Pseudodifferential Symbols .......................... 124
4.7 Integrable Hierarchies on the Poisson Lie Group
 int .................................................. 129
4.8 Bibliographical Notes ................................. 132
5 Double Loop and Elliptic Lie Groups ........................ 134
5.1 Central Extensions of Double Loop Groups and Their
Lie Algebras .......................................... 134
5.2 Coadjoint Orbits ...................................... 136
5.3 Holomorphic Loop Groups and Monodromy ................. 138
5.4 Digression: Definition of the Calogero-Moser
Systems ............................................... 142
5.5 The Trigonometric Calogero-Moser System and Affine
Lie Algebras .......................................... 146
5.6 The Elliptic Calogero-Moser System and Elliptic Lie
Algebras .............................................. 149
5.7 Bibliographical Notes ................................. 152
III Applications of Groups: Topological and Holomorphic
Gauge Theories ............................................ 155
1 Holomorphic Bundles and Hitchin Systems .................... 155
1.1 Basics on Holomorphic Bundles ......................... 155
1.2 Hitchin Systems ....................................... 159
1.3 Bibliographical Notes ................................. 162
2 Poisson Structures on Moduli Spaces ........................ 163
2.1 Moduli Spaces of Flat Connections on Riemann
Surfaces .............................................. 163
2.2 Poincare Residue and the Cauchy Stokes Formula ........ 170
2.3 Moduli Spaces of Holomorphic Bundles .................. 173
2.4 Bibliographical Notes ................................. 179
3 Around the Chern-Simons Functional ......................... 180
3.1 A Reminder on the Lagrangian Formalism ................ 180
3.2 The Topological Chern-Simons Action Functional ........ 184
3.3 The Holomorphic Chern-Simons Action Functional ........ 187
3.4 A Reminder on Linking Numbers ......................... 189
3.5 The Abelian Chern-Simons Path Integral and Linking
Numbers ............................................... 192
3.6 Bibliographical Notes ................................. 196
4 Polar Homology ............................................. 197
4.1 Introduction to Polar Homology ........................ 197
4.2 Polar Homology of Projective Varieties ................ 202
4.3 Polar Intersections and Linkings ...................... 206
4.4 Polar Homology for Affine Curves ...................... 209
4.5 Bibliographical Notes ................................. 211
Appendices .................................................... 213
A.l Root Systems .......................................... 213
1.1 Finite Root Systems .............................. 213
1.2 Semisimple Complex Lie Algebras .................. 215
1.3 Affine and Elliptic Root Systems ................. 216
1.4 Root Systems and Calogero Moser Hamiltonians ..... 218
A.2 Compact Lie Groups .................................... 221
2.1 The Structure of Compact Groups .................. 221
2.2 A Cohomology Generator for a Simple Compact
Group ............................................ 224
A.3 Krichever-Novikov Algebras ............................ 225
3.1 Holomorphic Vector Fields on * and the
Virasoro Algebra ................................. 225
3.2 Definition of the Krichever-Novikov Algebras
and Almost Grading ............................... 226
3.3 Central Extensions ............................... 228
3.4 Affine Krichever-Novikov Algebras, Coadjoint
Orbits, and Holomorphic Bundles .................. 231
A.4 Kahler Structures on the Virasoro and Loop Group
Coadjoint Orbits ...................................... 234
4.1 The Kahler Geometry of the Homogeneous Space
Diff(S1)/S1 ...................................... 234
4.2 The Action of Diff(S1) and Kahler Geometry
on the Based Loop Spaces ......................... 237
A.5 Diffeomorphism Groups and Optimal Mass Transport ..... 240
5.1 The Inviscid Burgers Equation as a Geodesic
Equation on the Diffeomorphism Group ............. 240
5.2 Metric on the Space of Densities and the Otto
Calculus ......................................... 244
5.3 The Hamiltonian Framework of the Riemannian
Submersion ....................................... 247
A.6 Metrics and Diameters of the Group of Hamiltonian
Diffeomorphisms ....................................... 250
6.1 The Hofer Metric and Bi-invariant Pseudometrics
on the Group of Hamiltonian Diffeomorphisms ...... 250
6.2 The Infinite L2-Diameter of the Group of
Hamiltonian Diffeomorphisms ...................... 252
A.7 Semidirect Extensions of the Diffeomorphism Group
and Gas Dynamics ...................................... 256
A.8 The Drinfeld Sokolov Reduction ........................ 260
8.1 The Drinfeld Sokolov Construction ................ 260
8.2 The Kupershmidt-Wilson Theorem and the Proofs .... 263
A.9 The Lie Algebra gl∞ ................................... 267
9.1 The Lie Algebra gl∞ and Its Subalgebras .......... 267
9.2 The Central Extension of gl∞ ..................... 268
9.3 g-Difference Operators and gl∞ ................... 269
A.10 Torus Actions on the Moduli Space of Flat
Connections ........................................... 272
10.1 Commuting Functions on the Moduli Space .......... 272
10.2 The Case of SU(2) ................................ 274
10.3 SL(n, ) and the Rational Ruijsenaars-Schneider
System ........................................... 277
References ................................................. 281
Index ......................................................... 301
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