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For a first orientation on the topics of the lecture, interested students can have a look at the following literature:
- Neeb, Karl-Hermann Monastir Lecture Notes on Infinite-Dimensional Lie Groups: Lecture notes for a summer school, aimed at giving an introduction into this topic. I will take major parts of the first part of the lecture from this text.
- Kriegl, Andreas and Michor, Peter W. The Convenient Setting of Global Analysis: A huge compendium on various aspects of infinite-dimensional manifolds and Lie-groups in the convenient setting. In many interesting cases, this setting coincides with the setting that we will use, so we may discuss some (very few) topics from this monograph.
Literature
The following literature list is a bit more detailed. However, many textbooks only treat finite-dimensional Lie group and may thus only serve as an overview of various topics of classical Lie theory.
-
H. Glöckner and K.-H. Neeb,
Infinite-dimensional Lie groups I+II
Graduate Texts in Mathematics
Springer-Verlag
in preparation
-
A. Kriegl and P. W. Michor,
The Convenient Setting of Global Analysis
Mathematical Surveys and Monographs
53
x+618 pages
American Mathematical Society
(1997)
-
W. Rudin,
Functional Analysis (second edition)
International Series in Pure and Applied Mathematics
xviii+424 pages
McGraw-Hill
(1991)
-
J. Hilgert and K.-H. Neeb,
Lie-Gruppen und Lie-Algebren (German)
361 pages
Vieweg-Verlag (Braunschweig
(1991)
-
H. Georgi,
Lie algebras in particle physics
Frontiers in Physics
54
xxii+255 pages
Benjamin/Cummings Publishing Co. Inc. Advanced Book Program
(1982)
From isospin to unified theories, With an introduction by Sheldon L. Glashow
-
G. L. Naber,
Topology, geometry, and gauge fields
Texts in Applied Mathematics
25
xviii+396 pages
Springer-Verlag
(1997)
Foundations
-
G. L. Naber,
Topology, geometry, and gauge fields
Applied Mathematical Sciences
141
xiv+443 pages
Springer-Verlag
(2000)
Interactions
-
R. W. Sharpe,
Differential geometry
Graduate Texts in Mathematics
166
xx+421 pages
Springer-Verlag
(1997)
Cartan's generalization of Klein's Erlangen program, With a foreword by S. S. Chern
-
S. Helgason,
Differential geometry, Lie groups, and symmetric spaces
Graduate Studies in Mathematics
34
xxvi+641 pages
American Mathematical Society
(2001)
Corrected reprint of the 1978 original
-
D. Bump,
Lie groups
Graduate Texts in Mathematics
225
xii+451 pages
Springer-Verlag
(2004)
-
A. L. Onishchik and E. B. Vinberg,
Lie groups and algebraic groups
Springer Series in Soviet Mathematics
xx+328 pages
Springer-Verlag
(1990)
Translated from the Russian and with a preface by D. A. Leites
Comments
-
I was asked to provide a reference for the fact that the Diffeomorphism group of a compact manifold may not be modelled on a Banach space in a reasonable way. The precise statement is that if a Banach-Lie group acts smoothly, effectively and transitively on a finite-dimensional manifold, then it automatically is finite-dimensional. This has been proven by Omori in
Hideki Omori,
On Banach-Lie groups acting on finite dimensional manifolds,
Tohoku Math. J. (2)
30
(2),
223-250,
1978
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