05835 - Structure and Geometry of Lie Groups [Hilgert-Neeb].pdf

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Springer Monographs
in
Mathematics
For further volumes:
www.springer.com/series/3733
Joachim Hilgert
r
Karl-Hermann Neeb
Structure and
Geometry
of Lie Groups
Joachim Hilgert
Mathematics Institute
University of Paderborn
Warburgerstr. 100
33095 Paderborn
Germany
hilgert@math.uni-paderborn.de
Karl-Hermann Neeb
Department of Mathematics
Friedrich-Alexander Universität
Erlangen-Nürnberg
Cauerstrasse 11
91054 Erlangen
Germany
neeb@mi.uni-erlangen.de
ISSN 1439-7382 Springer Monographs in Mathematics
ISBN 978-0-387-84793-1
e-ISBN 978-0-387-84794-8
DOI 10.1007/978-0-387-84794-8
Springer New York Dordrecht Heidelberg London
Library of Congress Control Number: 2011942060
Mathematics Subject Classification (2010): 7Bxx, 22Exx, 22Fxx
© Springer Science+Business Media, LLC 2012
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Preface
Nowadays there are plenty of textbooks on Lie groups to choose from, so we
feel we should explain why we decided to add another one to the row. Most
of the readily available books on Lie groups either aim at an elementary in-
troduction mostly restricted to matrix groups, or else they try to provide
the background on semisimple Lie groups needed in harmonic analysis and
unitary representation theory with as little general theory as possible. In
[HN91], we tried to exhibit the basic principles of Lie theory rather than
specific material, stressing the exponential function as the means of trans-
lating problems and solutions between the global and the infinitesimal level.
In that book, written in German for German students who typically do not
know differential geometry but are well versed in advanced linear algebra, we
avoided abstract differentiable manifolds by combining matrix groups with
covering arguments. Having introduced the basic principles, we demonstrated
their power by proving a number of standard and not so standard results on
the structure of Lie groups. The choice of results included owed a lot to
Hochschild’s book [Ho65], which even then was not so easy to come by.
This book builds on [HN91], but after twenty years of teaching and re-
search in Lie theory we found it indispensable to also have the differential
geometry of Lie groups available. Even though this is not apparent from the
text, the reason for this is the large number of applications and further devel-
opments of Lie theory in which differential manifolds are essential. Moreover,
we decided to include a number of structural results we found to be useful in
the past but not readily available in the textbook literature. The basic line
of thought now is:
•
•
•
•
Simple examples: Matrix groups
Tools from algebra: Lie algebras
Tools from geometry: Smooth manifolds
The basic principles: Lie groups, their Lie algebras, and the exponential
function
•
Structure theory: General Lie groups and special classes
•
Testing methods on examples: The topology of classical groups
•
A slight extension: Several connected components
While this book offers plenty of tested material for various introductory
courses such as
Matrix Groups, Lie Groups, Lie Algebras,
or
Differentiable

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