گروه‌های لی بی‌نهایت‌بعدی (ترجمه‌های مونوگرافی‌های ریاضی)

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. 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 Infinite-dimensional Calculus -- Infinite-dimensional Manifolds -- Infinite-dimensional Lie Groups -- Geometrical Structures On Orbits -- Fundamental Theorems For Differentiability -- Groups Of C{592} Diffeomorphisms On Compact Manifolds -- Linear Operators -- Several Subgroups Of D(m) -- Smooth Extension Theorems -- Group Of Diffeomorphisms On Cotangent Bundles -- Pseudodifferential Operators On Manifolds -- Lie Algebra Of Vector Fields -- Quantizations -- Poisson Manifolds And Quantum Groups -- Weyl Manifolds -- Infinite-dimensional Poisson Manifolds -- Appendices. Hideki Omori ; Translated By Hideki Omori. Includes Bibliographical References And Index. 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. The author 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.