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1.

図書
図書
1. Topological methods in hydrodynamics (: hbk)
Vladimir I. Arnold, Boris A. Khesin
出版情報: New York : Springer, c1998  xv, 374 p. ; 24 cm
シリーズ名: Applied mathematical sciences ; v. 125
2.

図書
図書
2. The geometry of infinite-dimensional groups (: hbk)
Boris Khesin, Robert Wendt
出版情報: Berlin : Springer, c2009  xii, 304 p. ; 24 cm
シリーズ名: Ergebnisse der Mathematik und ihrer Grenzgebiete ; 3. Folge . A series of modern surveys in mathematics ; v. 51
目次情報: 続きを見る
Preface
Introduction
Preliminaries / I：
Lie Groups and Lie Algebras / 1：
Lie Groups and an Infinite-Dimensional Setting / 1.1：
The Lie Algebra of a Lie Group / 1.2：
The Exponential Map / 1.3：
Abstract Lie Algebras / 1.4：
Adjoint and Coadjoint Orbits / 2：
The Adjoint Representation / 2.1：
The Coadjoint Representation / 2.2：
Central Extensions / 3：
Lie Algebra Central Extensions / 3.1：
Central Extensions of Lie Groups / 3.2：
The Euler Equations for Lie Groups / 4：
Poisson Structures on Manifolds / 4.1：
Hamiltonian Equations on the Dual of a Lie Algebra / 4.2：
A Riemannian Approach to the Euler Equations / 4.3：
Poisson Pairs and Bi-Hamiltonian Structures / 4.4：
Integrable Systems and the Liouville-Arnold Theorem / 4.5：
Symplectic Reduction / 5：
Hamiltonian Group Actions / 5.1：
Symplectic Quotients / 5.2：
Bibliographical Notes / 6：
Infinite-Dimensional Lie Groups: Their Geometry, Orbits, and Dynamical Systems / II：
Loop Groups and Affine Lie Algebras
The Central Extension of the Loop Lie algebra
Coadjoint Orbits of Affine Lie Groups
Construction of the Central Extension of the Loop Group
Diffeomorphisms of the Circle and the Virasoro-Bott Group
Coadjoint Orbits of the Group of Circle Diffeomorphisms
The Virasoro Coadjoint Action and Hill's Operators / 2.3：
The Virasoro-Bott Group and the Korteweg-de Vries Equation / 2.4：
The Bi-Hamiltonian Structure of the KdV Equation / 2.5：
Groups of Diffeomorphisms / 2.6：
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 / 3.3：
Semidirect Products: The Group Setting for an Ideal Magnetohydrodynamics and Compressible Fluids / 3.4：
Symplectic Structure on the Space of Knots and the Landau-Lifschitz Equation / 3.5：
Diffeomorphism Groups as Metric Spaces / 3.6：
The Group of Pseudodifferential Symbols / 3.7：
The Lie Algebra of Pseudodifferential Symbols
Outer Derivations and Central Extensions of [psi] DS
The Manin Triple of Pseudodifferential Symbols
The Lie Group of [alpha]-Pseudodifferential Symbols
The Exponential Map for Pseudodifferential Symbols
Poisson Structures on the Group of [alpha]-Pseudodifferential Symbols / 4.6：
Integrable Hierarchies on the Poisson Lie Group G[subscript INT] / 4.7：
Double Loop and Elliptic Lie Groups / 4.8：
Central Extensions of Double Loop Groups and Their Lie Algebras
Coadjoint Orbits
Holomorphic Loop Groups and Monodromy / 5.3：
Digression: Definition of the Calogero-Moser Systems / 5.4：
The Trigonometric Calogero-Moser System and Affine Lie Algebras / 5.5：
The Elliptic Calogero-Moser System and Elliptic Lie Algebras / 5.6：
Applications of Groups: Topological and Holomorphic Gauge Theories / 5.7：
Holomorphic Bundles and Hitchin Systems
Basics on Holomorphic Bundles
Hitchin Systems
Poisson Structures on Moduli Spaces
Moduli Spaces of Flat Connections on Riemann Surfaces
Poincare Residue and the Cauchy-Stokes Formula
Moduli Spaces of Holomorphic Bundles
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
Polar Homology
Introduction to Polar Homology
Polar Homology of Projective Varieties
Polar Intersections and Linkings
Polar Homology for Affine Curves
Appendices
Root Systems / A.1：
Finite Root Systems
Semisimple Complex Lie Algebras
Affine and Elliptic Root Systems
Root Systems and Calogero-Moser Hamiltonians
Compact Lie Groups / A.2：
The Structure of Compact Groups
A Cohomology Generator for a Simple Compact Group
Krichever-Novikov Algebras / A.3：
Holomorphic Vector Fields on [characters not reproducible] and the Virasoro Algebra
Definition of the Krichever-Novikov Algebras and Almost Grading
Affine Krichever-Novikov Algebras, Coadjoint Orbits, and Holomorphic Bundles
Kahler Structures on the Virasoro and Loop Group Coadjoint Orbits / A.4：
The Kahler Geometry of the Homogeneous Space Diff(S[superscript 1])/S[superscript 1]
The Action of Diff(S[superscript 1]) and Kahler Geometry on the Based Loop Spaces
Diffeomorphism Groups and Optimal Mass Transport / A.5：
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
Metrics and Diameters of the Group of Hamiltonian Diffeomorphisms / A.6：
The Hofer Metric and Bi-invariant Pseudometrics on the Group of Hamiltonian Diffeomorphisms / 6.1：
The Infinite L[superscript 2]-Diameter of the Group of Hamiltonian Diffeomorphisms / 6.2：
Semidirect Extensions of the Diffeomorphism Group and Gas Dynamics / A.7：
The Drinfeld-Sokolov Reduction / A.8：
The Drinfeld-Sokolov Construction / 8.1：
The Kupershmidt-Wilson Theorem and the Proofs / 8.2：
The Lie Algebra gl[subscript infinity] / A.9：
The Lie Algebra gl[subscript infinity] and Its Subalgebras / 9.1：
The Central Extension of gl[subscript infinity] / 9.2：
q-Difference Operators and gl[subscript infinity] / 9.3：
Torus Actions on the Moduli Space of Flat Connections / A.10：
Commuting Functions on the Moduli Space / 10.1：
The Case of SU(2) / 10.2：
SL(n, [characters not reproducible]) and the Rational Ruijsenaars-Schneider System / 10.3：
References
Index
Preface
Introduction
Preliminaries / I：
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