Pro-Lie groups which are infinite-dimensional Lie groups

Details Pro-Lie groups which are infinite-dimensional Lie groups Pro-Lie groups which are infinite-dimensional Lie groups

2006-09-25

arxiv.org/abs/math/0609684v1

Mathematics

Group Theory

Scientific paper

A pro-Lie group is a projective limit of a family of finite-dimensional Lie groups. In this note 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 neighborhood in the Lie algebra diffeomorphically onto an open identity neighborhood of the group.

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