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Journal of Lie Theory 21 (2011), No. 2, 347--383
Copyright Heldermann Verlag 2011

The Structure of Almost Connected Pro-Lie Groups

Karl H. Hofmann
Fachbereich Mathematik, Technische Universitšt, Schlossgartenstr. 7, 64289 Darmstadt, Germany

Sidney A. Morris
Graduate School of Information Technology and Mathematical Sciences, University of Ballarat, P. O. 663, Ballarat, Vic. 3353, Australia


\def\g{{\frak g}} \def\R{\mathbb{R}} \def\Z{\mathbb{Z}} \def\Aut{\mathop{\rm Aut}\nolimits} \def\Inn{\mathop{\rm Inn}\nolimits} Recalling that a topological group $G$ is said to be almost connected if the quotient group $G/G_0$ is compact, where $G_0$ is the connected component of the identity, we prove that for an almost connected pro-Lie group $G$, there exists a compact zero-dimens\-ional, that is, profinite, subgroup $D$ of $G$ such that $G=G_0D$. Further for such a group $G$, there are sets $I$, $J$, a compact connected semisimple group $S$, and a compact connected abelian group $A$ such that $G$ and $\R^I\times(\Z/2\Z)^J\times S\times A$ are homeomorphic. En route to this powerful structure theorem it is shown that the compact open topology makes the automorphism group $\Aut\g$ of a semisimple pro-Lie algebra $\g$ a topological group in which the identity component $(\Aut\g)_0$ is exactly the group $\Inn\g$ of inner automorphisms. In this situation, Inn(G) has a totally disconnected semidirect complement $\Delta$ such that $\Aut\g=(\Inn\g)\Delta$ and $\Aut\g/\Inn\g\cong \Delta$ as topological groups. The group $\Inn\g$ is a product of a family of connected simple centerfree Lie groups.

Keywords: Pro-Lie group, almost connected, maximal compact subgroup, conjugacy of subgroups, automorphism groups.

MSC: 22A05, 22D05, 22E10, 22E65

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