Infinite-Dimensional Lie Groups

430 Pages·1996·5.924 MB·English

by  Hideki Omori

Infinite -Dimensional
Lie Groups

Translations of
MATHEMATICAL
MONOGRAPHS
Volume 158
Infinite -Dimensional
Lie Groups
Hideki Omor!
Translated by
Hideki Omori
American Mathematical Society
y Providence, Rhode Island
Editorial Board
Shoshichi Kobayashi
Katsumi Nomizu (Chair)
MUGEN JIGEN RI GUNRON
(Infinite-dimensional Lie groups)
by Hideki Omori
Copyright © 1979 by Kinokuniya Co., Ltd.
Originally published in Japanese by Kinokuniya Co., Ltd., Tokyo, 1979
Translated from the Japanese by Hideki Omori
1991 Mathematics Subject Classification. Primary 58B25; Secondary 22E99, 81C25.
ABSTRACT. A general theory of infinite-dimensional Lie groups involving the implicit function
theorem and the Frobenius theorem is developed in this book. Related to the symbol calculus of
pseudodifferential operators, several noncommutative algebras such as Weyl algebras and algebras
of quantum groups are discussed in this English edition. The monograph is intended for research
mathematicians and graduate students.
Library of Congress Cataloging-in-Publication Data
Omori, Hideki, 1938-
[Mugen jigen RI gunron. English]
Infinite-dimensional Lie groups / Hideki Omori ; translated by Hideki Omori.
p. cm. - (Translations of mathematical monographs, ISSN 0065-9282 ; v. 158)
Includes bibliographical references and index.
ISBN 0-8218-4575-6 (alk. paper)
1. Infinite-dimensional manifolds. 2. Lie groups. I. Title. II. Series.
QA613.2.04613 1996
514'.223-dc20 96-38349
CIP
Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting
for them, are permitted to make fair use of the material, such as to copy a chapter for use
in teaching or research. Permission is granted to quote brief passages from this publication in
reviews, provided the customary acknowledgment of the source is given.
Republication, systematic copying, or multiple reproduction of any material in this publication
(including abstracts) is permitted only under license from the American Mathematical Society.
Requests for such permission should be addressed to the Assistant to the Publisher, American
Mathematical Society, P. O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also
be made by e-mail to [email protected].
© 1997 by the American Mathematical Society. All rights reserved.
Translation authorized by Kinokuniya Co., Ltd.
The American Mathematical Society retains all rights
except those granted to the United States Government.
Printed in the United States of America.
Q The paper used in this book is acid-free and falls within the guidelines
established to ensure permanence and durability.
10987654321 0100999897
Contents
Preface to the English Edition xi
Introduction 1
Chapter I. Infinite-Dimensional Calculus 5
§1 Topological linear spaces 5
§2 Integration 7
§3 Generalized Lie groups 9
§4 Rings and groups of linear mappings 14
§5 Definition of differentiable mappings 19
§6 Implicit function theorems 24
§7 Ordinary differential equations. Existence and regularity 27
§8 Examples of Sobolev chains 31
Chapter II. Infinite-Dimensional Manifolds 35
§1 F-manifolds, ILB-manifolds 35
§2 Vector bundles and affine connections 40
§3 Covariant exterior derivatives and Lie derivatives 43
§4 B-manifolds and gauge bundles 46
§5 Frobenius theorems 50
§6 ILH-manifolds and conformal structures 55
§7 Groups of bounded operators and Grassmann manifolds 58
Chapter III. Infinite-Dimensional Lie Groups 63
§1 Regular F-Lie groups 63
§2 Finite-dimensional subgroups, finite-codimensional subgroups 68
§3 Strong ILB-Lie groups 73
§4 Lie algebras, exponential mappings, subgroups 78
§5 Strong ILB-Lie groups are regular F-Lie groups 83
Chapter IV. Geometrical Structures on Orbits 91
§1 ILB-representations of strong ILB-Lie groups 91
§2 Geometrical structures defined by Lie algebras 95
§3 Structures given by elliptic complexes 99
§4 Several remarks 105
Chapter V. Fundamental Theorems for Differentiability 111
§ 1 Differential calculus on geodesic coordinate 111
§2 Bilateral ILB-chains and formal adjoint operators 116
§3 Differentiability and linear estimates 120
vii
viii CONTENTS
§4 Linear mappings of (E) into r(s(ir;E E)) 122
M
§5 Differentiability of compositions 125
§6 Continuity of the inverse 127
Chapter VI. Groups of C°° Diffeomorphisms on Compact Manifolds 133
§ 1 Invariant connections and Euler's equation of geodesic flows 133
§2 Groups of diffeomorphisms on compact manifolds 136
§3 Several subgroups of D(M) 140
§4 Subgroups of D(M) leaving a subset S invariant 142
§5 Remarks on global hypoellipticity 146
§6 Actions on differential forms 149
§7 Conjugacy of compact subgroups 154
Chapter VII. Linear Operators 159
§1 Operator valued holomorphic functions 159
§2 Spectra of compact operators 161
§3 Spectra of Hilbert-Schmidt operators 164
§4 Adjoint actions and the Hille-Yoshida theorem 167
§5 Elliptic differential operators 170
§6 Normed Lie algebras 178
Chapter VIII. Several Subgroups of D(M) 181
§1 The group Dd, (M) 181
§2 Multivalued volume forms 185
§3 Symplectic transformation groups 187
§4 Hamiltonian systems 191
§5 Contact algebras and Poisson algebras 195
§6 Contact transformations 198
§ 7 Deformation of a regular contact structure 201
Chapter IX. Smooth Extension Theorems 207
§1 Vector bundles and invariant homomorphisms 207
§2 Subbundles defined by invariant bundle homomorphisms 211
§3 The Frobenius theorem on strong ILB-Lie groups 215
§4 Elementary, smooth extension theorems on D(M) 217
§5 A Smooth extension theorem for differential operators 220
§6 The Frobenius theorem for finite codimensional Lie subalgebras 224
§7 The implicit function theorem via Frobenius theorem 226
§8 Existence of invariant connections and regularity of the exponential
mapping 229
Chapter X. Group of Diffeomorphisms on Cotangent Bundles 233
§1 Infinite-dimensional Lie algebras in general relativity 233
(T)/_m (TN)
§2 Strong ILH-Lie group with the Lie algebra 238
§3 Infinite-dimensional Lie groups with Lie algebra 1(TN) 240
§4 Regular F-Lie group with the Lie algebra (Tb) 244
§5 Groups of paths and loops 247
§6 Extensions by 2-cocycles 251
Chapter XI. Pseudodifferential Operators on Manifolds 255
CONTENTS ix
§1 Pseudodifferential operators on compact manifolds 255
§2 Products of pseudodifferential operators 259
§3 Several remarks on pseudodifferential operators 262
§4 Algebras and Lie algebras of pseudodifferential operators 266
§5 Fourier integral operators 272
Chapter XII. Lie Algebra of Vector Fields 277
§1 A generalization of the PS-theorem 277
§2 Orbits of Lie algebras 282
§3 Normal forms of vector fields 284
§4 The PS-theorem for Lie algebras leaving expansive subsets invari-
ant 288
Chapter XIII. Quantizations 293
§1 The correspondence principle 294
§2 Linear operators on Sobolev chains 296
§3 Quantized contact algebras 300
§4 Several algebraic tools 305
§5 Deformation quantization of Poisson algebras 309
§6 Several remarks and quantized Darboux theorem 314
Chapter XIV. Poisson Manifolds and Quan.tum Groups 319
§1 Examples of deformation quantized Poisson algebra 319
§2 Quantum groups 325
§3 Quantum SUQ (2), SUQ (1, 1) 328
§4 Deformation quantization of (S2, dV) 334
§5 Remarks on exact deformation quantizations 337
Chapter XV. Weyl Manifolds 341
§1 Weyl algebras, contact Weyl algebras 341
§2 Weyl functions 345
§3 Weyl diffeomorphisms 347
§4 Weyl manifolds 350
§5 Several structures on Weyl manifolds 355
Chapter XVI. Infinite-Dimensional Poisson Manifolds 361
§1 Equation of perfect fluid and geodesics 361
§2 Smooth functions on Sobolev chains 365
§3 Cotangent bundles of Sobolev manifolds 368
§4 Strong ILH-Lie groups as Sobolev manifolds 372
§5 The star-product on TG 374
Appendix I 381
Appendix II 389
Appendix III 395
References 403
Index 409