The Structure of Almost Connected Pro-Lie Groups

作者：Hofmann Karl H^*; Morris Sidney A

来源：Journal of Lie Theory, 2011, 21(2): 347-383.

• Pro-Lie group
• almost connected
• maximal compact subgroup
• conjugacy of subgroups

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-dimensional, that is, profinite, subgroup D of G such that G = G(0)D. Further for such a group G, there are sets I, J, a compact connected semisimple group S, and a compact connected abelia.n group A such that G and R-I x (Z/2Z)(J) x S x 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)Delta and Aut g/Inn g congruent to Delta as topological groups. The group Inn g is a product of a family of connected simple centerfree Lie groups.

• 出版日期2011