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Infinite-Dimensional Groups and Their Representations

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Book cover Lie Theory

Part of the book series: Progress in Mathematics ((PM,volume 228))

Summary

This article provides an introduction to the representation theory of Banach-Lie groups of operators on Hilbert spaces, where our main focus lies on highest weight representations and their geometric realization as spaces of holomorphic sections of a complex line bundle. After discussing the finite-dimensional case in Section I, we describe the algebraic side of the theory in Sections II and III. Then we turn in Sections IV and V to Banach-Lie groups and holomorphic representations of complex classical groups. The geometry of the coadjoint action is discussed in Section VI, and in the concluding Section VII all threads lead to a full discussion of the theory for the group U 2(H) of unitary operators u on a Hilbert space H for which u — 1 is Hilbert-Schmidt.

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Authors and Affiliations

  1. Technische Universität Darmstadt, Schlossgartenstrasse 7, D-64289, Darmstadt, Germany

    Karl-Hermann Neeb

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  1. Karl-Hermann Neeb
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Editors and Affiliations

  1. Laboratoire de Mathématiques (MAPMO), Université d’Orléans, F-45067, Orléans Cedex 2, France

    Jean-Philippe Anker

  2. Department of Mathematics and Computer Science, University of Southern Denmark, DK-5230, Odense M, Denmark

    Bent Orsted

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Neeb, KH. (2004). Infinite-Dimensional Groups and Their Representations. In: Anker, JP., Orsted, B. (eds) Lie Theory. Progress in Mathematics, vol 228. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-8192-0_2

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  • DOI: https://doi.org/10.1007/978-0-8176-8192-0_2

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4612-6483-5

  • Online ISBN: 978-0-8176-8192-0

  • eBook Packages: Springer Book Archive

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