The Structure of Almost Connected Pro-Lie Groups

Karl H. Hofmann¹ ; Sidney A. Morris²

¹Fachbereich Mathematik, Technische Universität, Schlossgartenstr. 7, 64289 Darmstadt, Germany
²Graduate School of Information Technology and Mathematical Sciences, University of Ballarat, P. O. 663, Ballarat, Vic. 3353, Australia

Journal of Lie Theory, Tome 21 (2011) no. 2, pp. 347-383

Résumé

\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.

Détail

Classification : 22A05, 22D05, 22E10, 22E65
Mots-clés : Pro-Lie group, almost connected, maximal compact subgroup, conjugacy of subgroups, automorphism groups