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

Source: [Pro-Lie groups which are infinite-dimensional Lie groups](<https://www.sota2.com/research/paper/68fa4d521e209131ce953994>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-25T09:14:37.176Z

Authors: K. H. Hofmann, K.-H. Neeb

Published: 2006-09-25T11:25:10.000Z

## Abstract

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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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/math/0609684>)

[PDF](<https://arxiv.org/pdf/math/0609684.pdf>)