Classical Hilbert–Lie groups, their extensions and their homotopy groups
Volume 55 / 2002
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.
