06609 - Infinite Dimensional Lie Transformations Groups [Omori].pdf

(3585 KB) Pobierz
Lecture Notes in
Mathematics
Edited by A. Dold and B, Eckmann
427
I
II
Hideki Omori
Infinite Dimensional
Lie Transformation~sl G rou ps
Springer-Verlag
Berlin.Heidelberg • New York 1974
Prof. Hideki Omori
Tokyo Metropolitan University
Fukazawa Setagaya
Tokyo/Japan
Library of Congress Cataloging in Publication Data
0mori, Hideki, 1938-
Infinite dimensional Lie transformation groups.
(LectUre notes in mathematics ; 427)
Bib!iogr ap~gy: p.
Includes index.
i. Transformation groups. 2. Lie groups.
3. Manifolds (Mathematics) I. Title. II. Series:
Lecture notes in mathematics (Berlin) ; 427.
QA3.L28 no. 427 [QA274.7]
512'.55
74-23625
AMS Subject Classifications (1 9?0): 54H15, 58 B99
ISBN 3-540-07013-3 Springer-Verlag Berlin • Heidelberg • New York
ISBN 0-387-07013-3 Springer-Verlag New York • Heidelberg • Berlin
This work is subject to copyright. All rights are reserved, whether the whole
or part of the material is concerned, specifically those of translation,
reprinting, re-use of illustrations, broadcasting, reproduction by photo-
copying machine or similar means, and storage in data banks.
Under § 54 of the German Copyright Law where copies are made for other
than private use, a fee is payable to the publisher, the amount of the fee to
be determined by agreement with the publisher.
© by Springer-Verlag Berlin • Heidelberg 1974. Printed in Germany.
Offsetdruck: Julius Beltz, Hemsbach/Bergstr.
Introduction
In this article,
dimensional
the author wants to discuss the possibilities
of an infinite
As such an
analogue of the theory of finite dimensional Lie groups.
analogue, we have already theories of Banach Lie groups or Hilbert Lie groups, which
are infinite dimensional
some suitable conditions
such groups.
(Cf.
analogues of finite dimensional
of simpleness,
linear groups, and under
table of
we have already a classification
[7] for instance.)
another theory of infinite dimensional Lie groups ?
:
Why does one want
Because of the following facts
a)
Let
G
be a Banach Lie group with Lie algebra
~.
If~
has no proper,
G
on a
finite codimensional
finite dimensional
ideal~ then the only possible smooth action of
is trivial.
(Cf. [33].)
smooth manlfold
The above fact shows that Banach Lie groups rarely act on finite dimensional
manifolds.
We do not have a single example of infinite dimensional
Banach Lie group
which acts effectively and transitively on a compact manifold.
Banaeh Lie groups which act on finite dimensional
dimensional
Lie groups.
manifolds
In many cases,
turn out to be finite
b)
If a Banaeh Lie group acts smoothly~
(i.e. leaves no foliation invariant)
effectively~
transitively and primitively
then it
on a finite dimensional manifold~
(Cf. [33],)
must be a finite dimensional Lie group.
In contrast,
Leslie
[20] showed that the group of all C~-diffeomorphisms
on a
closed manifold is a Frechet Lie group, namely this group is an infinite dimensional
manifold modeled on a Frechet space and the group operations
However,
are smooth.
For instance,
general Frechet manifolds are very difficult to treat.
in the definition of tangent bundles,
Of course,
there are some difficulties
hence in the
definition of the concept of C~-mappings.
function theorem nor a Frobenius
there is neither an implicit
Thus, it is difficult
to give
theorem in general.
XV
a theory of general Frechet Lie groups.
It is better to consider an intermediate
concept between Banach Lie groups and Frechet Lie groups.
For this purpose, the author defines the concept of strong ILB- and strong ILH-
Lie groups.
Even in this category, there is neither an implicit function theorem
However, we give a sufficient condition by
nor a Frobenius theorem in general.
using these concepts.
In
§I, the precise definition of strong ILB( or ILH)-Lie groups will be given,
In this category of groups, one can define
and some general facts will be discussed.
Lie algebras and exponential mappings.
Moreover, the following theorem holds :
Theorem A
The group structure of a strong ILB-LIe group is locally determined by
its Lie algebra. '
Now, it is natural to ask how many strong ILB ( or ILH)-Lie groups exist.
the main purpose of this article is to find examples.
We remark :
So,
The concept of
strong ILB ( or ILH)-Lie groups is something like the concept of structures defined
on topological groups.
structures.
A topological group can have many strong ILB-Lie group
Throughout this article, M
denotes a closed
M
with
C ~ - manifold and
C -topology.
~(M)
denotes
the group of all C -diffeomorphisms of
Theorem B
The topological group
~(M)
has both strong ILH- and strong ILB-Lie
gr,oup
structures.
By a slight modification of the proof of the above theorem, we get the following:
Theorem C
~j(M)
of
Suppose
~
(M)
M
has a C~-fiberin$ with a comRact fiber.
Then. the subgroup
which leaves the fibering ~
invariant has both strong ILH- and
ILB-Lie group structure~.
Theorem D
(M)
Let
K
be a compact subgroup in
~(M).
Then~ the subgrou~ @ K ( M )
k cK
of
of elements which commute pointwise with all
has both strong ILH- and

Plik z chomika:

Inne pliki z tego folderu:

Inne foldery tego chomika:

Zgłoś jeśli naruszono regulamin