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Axioms 2015, 4(3), 294-312; doi:10.3390/axioms4030294

Pro-Lie Groups: A Survey with Open Problems

1,2,†,* and 3,4,†,*
Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstrasse 7, Darmstadt 64289, Germany
Department of Mathematics, Tulane University, New Orleans, LA 70118, USA
Faculty of Science and Technology, Federation University Australia, Victoria 3353, Australia
School of Engineering and Mathematical Sciences, La Trobe University, Bundoora, Victoria 3086, Australia
These authors contributed equally to this work.
Authors to whom correspondence should be addressed.
Academic Editor: Angel Garrido
Received: 17 June 2015 / Revised: 15 July 2015 / Accepted: 17 July 2015 / Published: 24 July 2015
(This article belongs to the Special Issue Topological Groups: Yesterday, Today, Tomorrow)
View Full-Text   |   Download PDF [263 KB, uploaded 24 July 2015]


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. View Full-Text
Keywords: pro-Lie group; pro-Lie algebra; Lie group; Lie algebra; topological group; locally-compact group; unital topological algebra; exponential function; weakly-complete vector space pro-Lie group; pro-Lie algebra; Lie group; Lie algebra; topological group; locally-compact group; unital topological algebra; exponential function; weakly-complete vector space
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Hofmann, K.H.; Morris, S.A. Pro-Lie Groups: A Survey with Open Problems. Axioms 2015, 4, 294-312.

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