Regular infinite dimensional Lie groups

TL;DR

This paper explores the properties of regular infinite-dimensional Lie groups, demonstrating their geometric capabilities and implications for Lie algebra and group homomorphisms, extending finite-dimensional concepts.

Contribution

It shows that regular infinite-dimensional Lie groups enable advanced geometric structures and the integration of Lie algebra homomorphisms under certain conditions.

Findings

Parallel transport exists in regular Lie groups

Flat connections integrate to horizontal foliations

Lie algebra homomorphisms integrate to group homomorphisms under conditions

Abstract

Regular Lie groups are infinite dimensional Lie groups with the property that smooth curves in the Lie algebra integrate to smooth curves in the group in a smooth way (an `evolution operator' exists). Up to now all known smooth Lie groups are regular. We show in this paper that regular Lie groups allow to push surprisingly far the geometry of principal bundles: parallel transport exists and flat connections integrate to horizontal foliations as in finite dimensions. As consequences we obtain that Lie algebra homomorphisms intergrate to Lie group homomorphisms, if the source group is simply connected and the image group is regular.