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Infinite-dimensional Lie groups with a perspective to Mathematical Physics

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

 
  Seitenanfang  Impress 2009-11-05, Christoph Wockel