Lie theory for non-Lie groups

The class of locally compact groups admits a strong structure theory. This is due to the fact that—via approximation (projective limits)—important parts of the theory of Lie groups and Lie algebras carry over. This phenomenon becomes particularly striking if one assumes, in addition, that the groups under consideration are connected and of finite dimension. The aim of the present notes is to collect results and to show that Lie theory yields complete information about the rough structure (i.e., the lattice of closed connected subgroups) of locally compact finite-dimensional groups. Moreover, we shall describe the possibilities for locally compact connected non-Lie groups of finite dimension. We shall only consider Hausdorff groups (and shall, therefore, form quotients only with respect to closed subgroups—except in Example 1.7). The present notes grew out of an appendix to the author’s Habilitationsschrift [43]. The author gratefully acknowledges the influence and stimulation by Karl Heinrich Hofmann, Helmut Salzmann, Rainer Löwen, and Karl-Hermann Neeb.