**Translations of Mathematical Monographs**

1997;
415 pp;
Softcover

MSC: Primary 58;
Secondary 22; 81

Print ISBN: 978-1-4704-2635-4

Product Code: MMONO/158.S

List Price: $135.00

AMS Member Price: $108.00

MAA Member Price: $121.50

**Electronic ISBN: 978-1-4704-4573-7
Product Code: MMONO/158.E**

List Price: $135.00

AMS Member Price: $108.00

MAA Member Price: $121.50

# Infinite-Dimensional Lie Groups

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*Hideki Omori*

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.

#### Table of Contents

# Table of Contents

## Infinite-Dimensional Lie Groups

- Cover Cover11
- Title page v6
- Contents vii8
- Preface to the English edition xi12
- Introduction 114
- Chapter I. Infinite-dimensional calculus 518
- Chapter II. Infinite-dimensional manifolds 3548
- Chapter III. Infinite-dimensional Lie groups 6376
- Chapter IV. Geometric structures on orbits 91104
- Chapter V. Fundamental theorems for differentiability 111124
- Chapter VI. Groups of 𝐶^{∞} diffeomorphisms on compact manifolds 133146
- Chapter VII. Linear operators 159172
- Chapter VIII. Several subgroups of 𝒟(ℳ) 181194
- Chapter IX. Smooth extension theorems 207220
- Chapter X. Group of diffeomorphisms on cotangent bundles 233246
- Chapter XI. Pseudodifferential operators on manifolds 255268
- Chapter XII. Lie algebra of vector fields 277290
- Chapter XIII. Quantizations 293306
- Chapter XIV. Poisson manifolds and quantum groups 319332
- Chapter XV. Weyl manifolds 341354
- Chapter XVI. Infinite-dimensional Poisson manifolds 361374
- Appendix I 381394
- Appendix II 389402
- Appendix III 395408
- References 403416
- Index 409422
- Errata 417430
- Back Cover Back Cover1433