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Pro-Lie groups which are infinite-dimensional Lie groups

Published online by Cambridge University Press:  01 March 2009

K. H. HOFMANN  and
K.-H. NEEB
K. H. HOFMANN
Affiliation:
Technische Universität Darmstadt, Schlossgartenstrasse 7, 64289 Darmstadt, Germany. e-mail: [email protected], [email protected]
K.-H. NEEB
Affiliation:
Technische Universität Darmstadt, Schlossgartenstrasse 7, 64289 Darmstadt, Germany. e-mail: [email protected], [email protected]
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Abstract

A pro-Lie group is a projective limit of a family of finite-dimensional Lie groups. In this paper we show that a pro-Lie group G is a Lie group in the sense that its topology is compatible with a smooth manifold structure for which the group operations are smooth if and only if G is locally contractible. We also characterize the corresponding pro-Lie algebras in various ways. Furthermore, we characterize those pro-Lie groups which are locally exponential, that is, they are Lie groups with a smooth exponential function which maps a zero neighbourhood in the Lie algebra diffeomorphically onto an open identity neighbourhood of the group.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 146 , Issue 2 , March 2009 , pp. 351 - 378
Copyright
Copyright © Cambridge Philosophical Society 2008

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