Abstract
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 that 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 the 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 that links the two. (A topological vector space is weakly complete if it is isomorphic to a power RX 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 to pro-Lie groups.
Highlights
In 1900, David Hilbert presented a seminal address to the International Congress of Mathematicians inParis
In order to understand the structure of pro-Lie groups in our sense, we developed a highly infinite-dimensional Lie theory of considerable complexity
It has been an ongoing effort to replace Rn by more general, possibly infinite-dimensional, topological vector spaces supporting differentiable structures. The most advanced such theory is the theory of Lie groups on differentiable or smooth manifolds modeled on open subsets of locally-convex vector spaces and their real analysis as used by Helge Glöckner and Karl-Hermann Neeb
Summary
He initiated a program by formulating 23 problems, which influenced a vast amount of research of the 20th century The fifth of these problems asked whether every locally-Euclidean topological group admits a Lie group structure, see [1]. This motivated an enormous volume of work on locally-compact groups during the first half of the 20th century. It has been an ongoing effort to replace Rn by more general, possibly infinite-dimensional, topological vector spaces supporting differentiable structures The most advanced such theory is the theory of Lie groups on differentiable or smooth manifolds modeled on open subsets of locally-convex vector spaces and their real analysis as used by Helge Glöckner and Karl-Hermann Neeb.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
