The structure of abelian pro-Lie groups
Identifieur interne : 000D71 ( Main/Exploration ); précédent : 000D70; suivant : 000D72The structure of abelian pro-Lie groups
Auteurs : Karl H. Hofmann [Allemagne] ; Sidney A. Morris [Australie]Source :
- Mathematische Zeitschrift [ 0025-5874 ] ; 2004-12-01.
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Abstract
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.
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DOI: 10.1007/s00209-004-0685-5
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Hofmann</name><affiliation wicri:level="3"><country xml:lang="fr">Allemagne</country><wicri:regionArea>Fachbereich Mathematik, Darmstadt University of Technology, Schlossgartenstrasse 7, 64289, Darmstadt</wicri:regionArea><placeName><region type="land" nuts="1">Hesse (Land)</region><region type="district" nuts="2">District de Darmstadt</region><settlement type="city">Darmstadt</settlement></placeName></affiliation><affiliation wicri:level="1"><country wicri:rule="url">Allemagne</country></affiliation></author><author><name sortKey="Morris, Sidney A" sort="Morris, Sidney A" uniqKey="Morris S" first="Sidney A." last="Morris">Sidney A. 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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.</div></front></TEI><affiliations><list><country><li>Allemagne</li><li>Australie</li></country><region><li>District de Darmstadt</li><li>Hesse (Land)</li></region><settlement><li>Darmstadt</li></settlement></list><tree><country name="Allemagne"><region name="Hesse (Land)"><name sortKey="Hofmann, Karl H" sort="Hofmann, Karl H" uniqKey="Hofmann K" first="Karl H." last="Hofmann">Karl H. Hofmann</name></region><name sortKey="Hofmann, Karl H" sort="Hofmann, Karl H" uniqKey="Hofmann K" first="Karl H." last="Hofmann">Karl H. Hofmann</name></country><country name="Australie"><noRegion><name sortKey="Morris, Sidney A" sort="Morris, Sidney A" uniqKey="Morris S" first="Sidney A." last="Morris">Sidney A. Morris</name></noRegion><name sortKey="Morris, Sidney A" sort="Morris, Sidney A" uniqKey="Morris S" first="Sidney A." last="Morris">Sidney A. Morris</name></country></tree></affiliations></record>Pour manipuler ce document sous Unix (Dilib)
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