Infinite-dimensional Lie groups / Hideki Omori ; translated by Hideki Omori.

Includes bibliographical references (p. 403-407) and index.

Chapters
Introduction
Chapter I. Infinite-dimensional calculus
Chapter II. Infinite-dimensional manifolds
Chapter III. Infinite-dimensional Lie groups
Chapter IV. Geometric structures on orbits
Chapter V. Fundamental theorems for differentiability
Chapter VI. Groups of C∞
diffeomorphisms on compact manifolds
Chapter VII. Linear operators
Chapter VIII. Several subgroups of D(M)
Chapter IX. Smooth extension theorems
Chapter X. Group of diffeomorphisms on cotangent bundles
Chapter XI. Pseudodifferential operators on manifolds
Chapter XII. Lie algebra of vector fields
Chapter XIII. Quantizations
Chapter XIV. Poisson manifolds and quantum groups
Chapter XV. Weyl manifolds
Chapter XVI. Infinite-dimensional Poisson manifolds

Translations of Mathematical Monographs
Volume: 158; 1997; 415 pp
MSC: Primary 58; Secondary 22; 81;
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.