Pro-Lie Groups: A survey with Open Problems

A topological group is called a pro-Lie group if it is isomorphic to a closed subgroup of a product of finite-dimensional real Lie groups. This class of groups is closed under the formation of arbitrary products and closed subgroups and forms a complete category. It includes each finite-dimensional Lie group, each locally compact group which has a compact quotient group modulo its identity component and thus, in particular, each compact and each connected locally compact group; it also includes all locally compact abelian groups. This paper provides an overview of the structure theory and Lie theory of pro-Lie groups including results more recent than those in the authors' reference book on pro-Lie groups. Significantly, it also includes a review of the recent insight that weakly complete unital algebras provide a natural habitat for both pro-Lie algebras and pro-Lie groups, indeed for the exponential function which links the two. (A topological vector space is weakly complete if it is isomorphic to a power $\R^X$ of an arbitrary set of copies of $\R$. This class of real vector spaces is at the basis of the Lie theory of pro-Lie groups.) The article also lists 12 open questions connected with pro-Lie groups.