- Title
- The structure of abelian pro-Lie groups
- Creator
- Hofmann, Karl; Morris, Sidney
- Date
- 2004
- Type
- Text; Journal article
- Identifier
- http://researchonline.federation.edu.au/vital/access/HandleResolver/1959.17/55325
- Identifier
- vital:408
- Identifier
-
https://doi.org/10.1007/s00209-004-0685-5
- Identifier
- ISSN:0025-5874
- Abstract
- A pro-Lie group is a projective limit of a projective system of finite dimensional Lie groups. A prodiscrete group is a complete abelian topological group in which the open normal subgroups form a basis of the filter of identity neighborhoods. It is shown here that an abelian pro-Lie group is a product of (in general infinitely many) copies of the additive topological group of reals and of an abelian pro-Lie group of a special type; this last factor has a compact connected component, and a characteristic closed subgroup which is a union of all compact subgroups; the factor group modulo this subgroup is pro-discrete and free of nonsingleton compact subgroups. Accordingly, a connected abelian pro-Lie group is a product of a family of copies of the reals and a compact connected abelian group. A topological group is called compactly generated if it is algebraically generated by a compact subset, and a group is called almost connected if the factor group modulo its identity component is compact. It is further shown that a compactly generated abelian pro-Lie group has a characteristic almost connected locally compact subgroup which is a product of a finite number of copies of the reals and a compact abelian group such that the factor group modulo this characteristic subgroup is a compactly generated prodiscrete group without nontrivial compact subgroups.; C1
- Publisher
- Springer
- Relation
- Mathematische Zeitschrift Vol. 248, no. 4 (Dec 2004), p. 867-891
- Rights
- Copyright Springer
- Rights
- This metadata is freely available under a CCO license
- Subject
- 0101 Pure Mathematics; Abelian topological group; Projective limit; Lie group; Exponential function; Locally compact group; Vector subgroup
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