A+ CATEGORY SCIENTIFIC UNIT

Classical Hilbert–Lie groups, their extensions and their homotopy groups

Volume 55 / 2002

Karl-Hermann Neeb Banach Center Publications 55 (2002), 87-151 MSC: Primary 22E65; Secondary 57T20. DOI: 10.4064/bc55-0-6

Abstract

Let $H$ be a complex Hilbert space and $D$ a hermitian operator on $H$ with finite spectrum. Then the operators for which the commutator with $D$ is of Schatten class $p$ form a Banach algebra $B_p(H,D)$. In the present paper we study groups $\operatorname{GL}_p(H,D)$ associated to this kind of Lie algebra, and also groups $\operatorname{GL}_p(H,I,D)$ associated to the Lie subalgebras $B_p(H,I,D) := \{x \in B_p(H,D) \colon Ix^*I^{-1} =- x\}$, where $I$ is an antilinear isometry with $I^2 \in \{\pm {\bf1}\}$. For $p = 2$ we determine the full second continuous cohomology for these Lie algebras, and for the groups we compute all homotopy groups. These results then lead to a direct description of universal central extensions of the groups $\operatorname{GL}_2(H,D)$, $\operatorname{GL}_2(H,I,D)$ and some of their real forms. In particular we obtain the infinite-dimensional metaplectic and metagonal groups as special examples. In a last section we discuss associated complex flag manifolds and show that the unitary forms of the complex groups act transitively.

Authors

  • Karl-Hermann NeebFachbereich Mathematik
    Darmstadt University of Technology
    Schlossgartenstrasse 7
    64289 Darmstadt, Germany
    e-mail

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