Pro-Lie Groups: A Survey with Open Problems
AbstractA 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
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Hofmann, K.H.; Morris, S.A. Pro-Lie Groups: A Survey with Open Problems. Axioms 2015, 4, 294-312.
Hofmann KH, Morris SA. Pro-Lie Groups: A Survey with Open Problems. Axioms. 2015; 4(3):294-312.Chicago/Turabian Style
Hofmann, Karl H.; Morris, Sidney A. 2015. "Pro-Lie Groups: A Survey with Open Problems." Axioms 4, no. 3: 294-312.