Infinite-Dimensional Lie Groups

by 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.

415 pages, Hardcover

First published February 1, 1997

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II.7 Groups of bounded operators and Grassmann manifolds. (page 59)

If A is an associative B-algebra then regarding A as a Lie algebra by the commutator bracket we see that the set of all linear invertible elements A^x is the B-Lie group with Lie algebra A.