Infinite-Dimensional Lie Groups

Cover

S Title

Infinite -Dimensional Lie Groups

© 1997 by the American Mathematical Society
     ISBN 0-8218-4575-6
     QA613.2.04613 1996 514'.223-dc20
     LCCN 96038349

Contents

Preface to the English Edition

Introduction

CHAPTER I  Infinite-Dimensional Calculus

     §I.1. Topological linear spaces

     §I.2. Integration

     §I.3. Generalized Lie groups

     §I.4. Rings and groups of linear mappings

     §I.5. Definition of differentiable mappings

     §I.6. Implicit function theorems

     §I.7. Ordinary differential equations. Existence and regularity

     §I.8. Examples of Sobolev chains

CHAPTER II  Infinite-Dimensional Manifolds

     §II.1. F-manifolds, ILB-manifolds

     §II.2. Vector bundles and affine connections

     §II.3. Covariant exterior derivatives and Lie derivatives

     §II.4. B-manifolds and gauge bundles

     §II.5. Frobenius theorems

     §II.6. ILH-manifolds and conformal structures

     §II.7 Groups of bounded operators and Grassmann manifolds

CHAPTER III  Infinite-Dimensional Lie Groups

     §III.1. Regular F-Lie groups

     §III.2. Finite-dimensional subgroups, finite-codimensional subgroups

     §III.3. Strong ILB-Lie groups

     §III.4. Lie algebras, exponential mappings, subgroups

     §III.5. Strong ILB-Lie groups are regular F-Lie groups

CHAPTER IV  Geometric Structures on Orbits

     §IV.1. ILB-representations of strong ILB-Lie groups

     §IV.2. Geometrical structures defined by Lie algebras

     §IV.3. Structures given by elliptic complexes

     §IV.4. Several remarks

CHAPTER V  Fundamental Theorems for Differentiability

     §V.1. Differential calculus using geodesic coordinates

     §V.2. Bilateral ILB-chains and formal adjoint operators

     §V.3. Differentiability and linear estimates

     §V.4. Linear mappings of l(E) into l(S(lrTME))

     §V.5. Differentiability of compositions

     §V.6. Continuity of the inverse

CHAPTER VI  Groups of C°° Diffeomorphisms on Compact Manifolds

     §VI.1. Invariant connections and Euler's equation of geodesic flows

     §VI.2. Groups of diffeomorphisms on compact manifolds

     §VI.3. Several subgroups of D(M)

     §VI.4. Subgroups of D(M) leaving a subset S invari

     §VI.5. Remarks on global hypoellipticity

     §VI.6. Actions on differential forms

     §VI.7. Conjugacy of compact subgroups

CHAPTER VII  Linear Operators

     §VII.1. Operator valued holomorphic functions

     §VII.2. Spectra of compact operators

     §VII.3. Spectra of Hilbert-Schmidt operators

     §VII.4. Adjoint actions and the Hille-Yoshida theorem

     §VII.5. Elliptic differential operators

     §VII.6. Normed Lie algebras

CHAPTER VIII  Several Subgroups of D(M)

     §VIII.1. The group Dd (M)

     §VIII.2. Multivalued volume forms

     §VIII.3. Symplectic transformation groups

     §VIII.4. Hamiltonian systems

     §VIII.5. Contact algebras and Poisson algebras

     §VIII.6. Contact transformations

     §VIII.7. Deformation of a regular contact structure

CHAPTER IX  Smooth Extension Theorems

     §IX.1. Vector bundles and invariant homomorphisms

     §IX.2. Subbundles defined by invariant bundle homomorphisms

     §IX.3. The Frobenius theorem on strong ILB-Lie groups

     §IX.4. Elementary, smooth extension theorems on D(M)

     §IX.6. The Frobenius theorem for finite-codimensional subalgebra

     §IX.7. The implicit function theorem via Frobenius theorems

     §IX.8. Existence of invariant connections and regularity of the exponential mapping

CHAPTER X  The Group of Diffeomorphisms on Cotangent Bundles

     §X.1. Infinite-dimensional Lie algebras in general relativity

     §X.2. Strong ILH-Lie groups with the Lie algebra E 1(TN) lE -'n-1(TN )

     §X.3. Infinite-dimensional Lie groups with Lie algebra El (TN)

     §X.4. Regular F-Lie group with the Lie algebra

     §X.5. Groups of paths and loops

     §X.6. Extensions by 2-cocycles

CHAPTER XI  Pseudodifferential Operators on Manifolds

     §XI.1. Pseudodifferential operators on compact manifolds

     §XI.2. Products of pseudodifferential operators

     §XI.3. Several remarks on pseudodifferential operators

     §XI.4. Algebras and Lie algebras of pseudodifferential operators

     §XI.5. Fourier integral operators

CHAPTER XII  Lie Algebra of Vector Fields

     §XII.1. A generalization of the PS-theorem

     §XII.2. Orbits of Lie algebras

     §XII.3. Normal forms of vector fields

     §XII.4. The PS-theorem for Lie algebras leaving expansive subsets invariant

CHAPTER XIII  Quantizations

     §XIII.1. The correspondence principle

     §XIII.2. Linear operators on Sobolev chains

     §XIII.3. Quantized contact algebras

     §XIII.4. Several algebraic tools

     §XIII.5. Deformation quantization of Poisson algebras

     §XIII.6. Several remarks and the quantized Darboux theorem

CHAPTER XIV  Poisson Manifolds and Quantum Groups

     §XIV.1. Examples of deformation quantized Poisson algebras

     §XIV.2. Quantum groups

     §XIV.3. Quantum SUq (2), SUq (1, 1)

     §XIV.4. Deformation quantization of (S2, dV )

     §XIV.5. Remarks on exact deformation quantizations

CHAPTER XV  Weyl Manifolds

     §XV.1. Weyl algebras, contact Weyl algebras

     §XV.2. Weyl functions

     §XV.3. Weyl diffeomorphisms

     §XV.4. Weyl manifolds

     §XV.5. Several structures on Weyl manifolds

CHAPTER XVI  Infinite-Dimensional Poisson Manifolds

     §XVI.1. Equation of perfect fluid and geodesics

     §XVI.2. Smooth functions on Sobolev chains

     §XVI.3. Cotangent bundles of Sobolev manifolds

     §XVI.4. Strong ILH-Lie groups as Sobolev manifolds

     §XVI.5. The star-product on TG

APPENDIX I
     Proof of §VI, Theorem 1.4

APPENDIX II

     Consistency conditions

APPENDIX III
     Construction of \pi

References

Index