Table Of ContentI n f i n i te Dimensional
in Geometry and
Representation Theory
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I n f i n i te D i m e n s i o n al
L
ife rfwPfffc.
le Groups
in Geometry and
Representation Theory
Washington, DC, USA
17-21 August 2000
Editors
Augustia Banyaga
Pennsylvania Stale University, USA
Joshua A Leslie
Howard University, USA
Thierry Robart
Howard University, USA
V|fe World Scientific
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Published by
World Scientific Publishing Co. Pte. Ltd.
P O Box 128, Fairer Road, Singapore 912805
USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
INFINITE DIMENSIONAL LIE GROUPS IN GEOMETRY AND
REPRESENTATION THEORY
Copyright © 2002 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
electronic or mechanical, including photocopying, recording or any information storage and retrieval
system now known or to be invented, without written permission from the Publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright
Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to
photocopy is not required from the publisher.
ISBN 981-238-068-X
Printed in Singapore by Uto-Print
Preface
This volume contains papers delivered at the occasion of the 2000 Howard
fest on Infinite Dimensional Lie Groups in Geometry and Representation The
ory. The five day International Conference was held on the main campus of
Howard University from the 17th to the 21st August, 2000. We believe that
the collected papers, by presenting important recent developments, should
offer a valuable source of inspiration for advanced graduate students and/or
established researchers in the field. All papers have been refereed.
A Short overview:
The book opens with a topological characterization of regular Lie groups
in the context of Lipschitz-metrizable groups, a class that contains all strong
ILB-Lie groups introduced by Omori in the early seventies (Josef Teichmann).
It then treats the integrability problem of various important infinite dimen
sional Lie algebras: a canonical approach of the general integration problem
based on charts of the second kind and illustrated with the isotropy group of
local analytic vector fields is described (Thierry Robart); a result of Good
man and Wallach is extented to a very large class of Kac-Moody algebras
associated to generalized symmetrizable Cartan matrices (Joshua Leslie); and
within the framework of bounded geometry the known Lie group structure of
invertible Fourier integral operators on compact manifolds is shown to hold
equally for open manifolds (Rudolf Schmid).
The volume contains also important contributions at the forefront of mod
ern geometry. Firstly, the main properties of Leibniz algebroids are studied
here (Aissa Wade). The concept of Leibniz algebroid was introduced recently
in the study of Nambu-Poisson structures. As weakened version of that of Lie
algebroid, it represents a far-reaching generalization of the classical concept
of Lie algebra.
There are five papers devoted to locally conformal symplectic geome
try (Augustin Banyaga, Stefan Haller), contact geometry (Philippe Rukim-
bira), smooth orbifold structures (Joseph E. Borzellino and Victor Brunsden)
and the equivalence problem of Poisson and symplectic structures (Augustin
Banyaga and Paul Donato). There the focus is mainly on the interaction
between the studied structures and their associated infinite dimensional Lie
groups of symmetries in the spirit of the 1872 Erlanger Programme of Fe
lix Klein. It is shown in particular that the automorphism groups of locally
conformal symplectic structures and of smooth orbifolds determine the corre
sponding structures. These strong results have many applications. They can
v
vi
be used among other for classification purpose; for instance locally confor-
mal symplectic structures are classified according to a certain homomorphism
(the Lee homomorphism) on their automorphism groups. Unit Reeb fields on
contact manifolds, viewed as maps from the manifold into its unit tangent
bundle, are characterized as harmonic maps or minimal embeddings under
certain conditions.
The book concludes with penetrating remarks concerning the concept of
amenability, infinite dimensional groups and representation theory (Vladimir
Pestov).
List of Participants/Authors
Augustin Banyaga (Penn State University), Joe Borzellino (California
State Polytechnic University), Michel Boyom (Universite de Montpellier II,
France), Victor Brunsden (Penn State University), Paul Donato (Centre de
Mathematiques et d'Informatique, Marseille, France), Stefan Haller (Uni
versity of Vienna, Austria), Patrick Iglesias (Centre de Mathematiques et
d'Informatique, Marseille, France), Joshua Leslie (Howard University), Pe
ter Michor (Vienna University, Austria), Hideki Omori (Science University
of Tokyo, Japan), Vladimir Pestov (Victoria University of Wellington, New
Zealand), Tudor Ratiu (Ecole Polytechnique Federale de Lausanne, Switzer
land), Thierry Robart (Howard University), Philippe Rukimbira (Florida In
ternational University), Rudolf Schmid (Emory University), Josef Teichmann
(Technische Universitat Wien, Austria), Aissa Wade (Penn State University).
Contributions
• Inheritance properties for Lipschitz-metrizable Frolicher groups by Josef
Teichmann,
• Around the exponential mapping by Thierry Robart,
• On a solution to a global inverse problem with respect to certain general
ized symmetrizable Kac-Moody Lie algebras by Joshua Leslie,
• The Lie group of Fourier integral operators on open manifolds by Rudolf
Schmid,
• On Some properties of Leibniz Algebroids by Aissa Wade,
• On the geometry of locally conformal symplectic manifolds by Augustin
Banyaga,
vii
• Some properties of locally conformal symplectic manifolds by Stefan
Haller,
• Criticality of unit contact vector fields by Philippe Rukimbira,
• Orbifold Homeomorphism and Diffeomorphism Groups by Joseph E.
Borzellino and Victor Brunsden,
• A note on Isotopies of Symplectic and Poisson Structures by Augustin
Banyaga and Paul Donato,
• Remarks on actions on compacta by some infinite-dimensional groups by
Vladimir Pestov,
Acknowledgment
We would like to express our deepest gratitude to the National Security
Agency. This Conference wouldn't have been possible without its generous
support.
We also wish to thank all the participants, authors, referees and colleagues
for their various and irreplaceable contributions, specially Aissa Wade and
Philippe Rukimbira for their constant help during the preparation of this
volume.
A. Banyaga, J. Leslie & T. Robart
May 2, 2002
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Contents
Inheritance Properties for Lipschitz-Metrizable Frolicher Groups 1
J. Teichmann
Around the Exponential Mapping 11
T. Robart
On a Solution to a Global Inverse Problem with Respect to Certain
Generalized Symmetrizable Kac-Moody Algebras 31
J. A. Leslie
On Some Properties of Leibniz Algebroids 65
A. Wade
On the Geometry of Locally Conformal Symplectic Manifolds 79
A. Banyaga
Some Properties of Locally Conformal Symplectic Manifolds 92
S. Holler
Criticality of Unit Contact Vector Fields 105
P. Rukimbira
Orbifold Homeomorphism and Diffeomorphism Groups 116
J. E. Borzellino & V. Brunsden
A Note on Isotopies of Symplectic and Poisson Structures 138
A. Banyaga & P. Donato
Remarks on Actions on Compacta by Some Infinite-Dimensional
Groups 145
V. Pestov
IX
INHERITANCE PROPERTIES FOR
LIPSCHITZ-METRIZABLE FROLICHER GROUPS
JOSEF TEICHMANN
Institute of financial and actuarial mathematics, Technical University of Vienna,
Wiedner Hauptstrafie 8-10, A-1040 Vienna, Austria
E-mail: josef teichmann@fam. tuwien. ac. at
Prolicher groups, where the notion of smooth map makes sense, are introduced. On
Prolicher groups we can formulate the concept of Lipschitz metrics. The resulting
setting of Prolicher-Lie groups can be compared to generalized Lie groups in the
sense of Hideki Omori. Furthermore Lipschitz-metrics on Prolicher groups allow
to prove convergence of approximation schemes for differential equations on Lie
groups. We prove several inheritance properties for Lipschitz metrics.
1 Introduction
Lipschitz-metrizable groups have been introduced in 6 to show that regularity
of Lie groups (see 1 for all necessary details on Lie groups) is closely connected
to some approximation procedures possible on Lie groups. The convergence
of these approximation schemes is guaranteed by Lipschitz metrics.
In the work of Hideki Omori et al. the beautiful framework of strong ILB-
Lie groups is provided (see 2 for example), where the problem of regularity is
solved by analytic assumption on the group-multiplication in the charts. The
advantage of Lipschitz-metrizable groups is that the notion is "inner", i.e.
formulated on the Lie group itself without charts. All strong 7L5-groups are
Lipschitz-metrizable regular groups (see 6, Corollary 2.10). Given a Lipschitz
metrizable Lie group we can - by approximation schemes - characterize the
existence of exponential and evolution maps. This can also be applied to
solve more general equations as stochastic differential equations on Lipschitz-
metrizable Lie groups.
In this note we motivate the method of Lipschitz metrics and investi
gate roughly the inheritance properties of Lipschitz-metrizable Frolicher-Lie
groups.
Definition 1. A non-empty set X, a set of curves Cx C Map(M.,X) and a
set of mappings Fx C Map(X, E.) are called a Frolicher space if the following
conditions are satisfied:
1. A map f : X -> R belongs to F if and only if f o c € C°°(R,R) for
x
c£C.
x
1
