This book develops, from the viewpoint of abstract group theory, a general theory of infinite-dimensional Lie groups involving the implicit function theorem and the Frobenius theorem. Omori treats as infinite-dimensional Lie groups all the real, primitive, infinite transformation groups studied by E. Cartan. The book discusses several noncommutative algebras such as Weyl algebras and algebras of quantum groups and their automorphism groups. The notion of a noncommutative manifold is described, and the deformation quantization of certain algebras is discussed from the viewpoint of Lie algebras.
This edition is a revised version of the book of the same title published in Japanese in 1979.
Readership: Graduate students, research mathematicians, mathematical physicists and theoretical physicists interested in global analysis and on manifolds.
Author(s): Hideki Omori
Series: Translations of Mathematical Monographs, Vol. 158
Edition: Revised
Publisher: American Mathematical Society
Year: 1996
Language: English
Pages: C, xii+415, B
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
