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Infinite-Dimensional Lie Groups
Hideki Omori, Science University of Tokyo

Translations of Mathematical Monographs
1997; 415 pp; hardcover
Volume: 158
ISBN-10: 0-8218-4575-6
ISBN-13: 978-0-8218-4575-2
List Price: US$135
Member Price: US$108
Order Code: MMONO/158
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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.


Graduate students, research mathematicians, mathematical physicists and theoretical physicists interested in global analysis and on manifolds.

Table of Contents

  • Introduction
  • Infinite-dimensional calculus (Chapter I)
  • Infinite-dimensional manifolds (Chapter II)
  • Infinite-dimensional Lie groups (Chapter III)
  • Geometric structures on orbits (Chapter IV)
  • Fundamental theorems for differentiability (Chapter V)
  • Groups of \(C^\infty\) diffeomorphisms on compact manifolds (Chapter VI)
  • Linear operators (Chapter VII)
  • Several subgroups of \({\scr D}(M)\) (Chapter VIII)
  • Smooth extension theorems (Chapter IX)
  • The group of diffeomorphisms on cotangent bundles (Chapter X)
  • Pseudodifferential operators on manifolds (Chapter XI)
  • Lie algebra of vector fields (Chapter XII)
  • Quantizations (Chapter XIII)
  • Poisson manifolds and quantum groups (Chapter XIV)
  • Weyl manifolds (Chapter XV)
  • Infinite-dimensional Poisson manifolds (Chapter XVI)
  • Appendix I
  • Appendix II
  • Appendix III
  • References
  • Index
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