The Structure of Almost Connected Pro-Lie Groups

摘要

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 con-nected component of the identity, we prove that for an almost connected pro-Lie group G, there exists a compact zero-dimensional, 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 × (Z/2Z)~J × S × 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 A such that Aut g = (Inn g)Δ and Aut g/ Inn g Δ as topological groups. The group Inn g is a product of a family of connected simple centerfree Lie groups.