
Journal of Lie Theory 21 (2011), No. 2, 347383 Copyright Heldermann Verlag 2011 The Structure of Almost Connected ProLie Groups Karl H. Hofmann Fachbereich Mathematik, Technische Universität, Schlossgartenstr. 7, 64289 Darmstadt, Germany hofmann@mathematik.tudarmstadt.de Sidney A. Morris Graduate School of Information Technology and Mathematical Sciences, University of Ballarat, P. O. 663, Ballarat, Vic. 3353, Australia morris.sidney@gmail.com [Abstractpdf] \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 proLie group $G$, there exists a compact zerodimens\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 proLie 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: ProLie group, almost connected, maximal compact subgroup, conjugacy of subgroups, automorphism groups. MSC: 22A05, 22D05, 22E10, 22E65 [ Fulltextpdf (450 KB)] for subscribers only. 