# Direct limits of regular Lie groups

## TL;DR

This paper establishes conditions under which a regular Lie group formed as a union of smaller regular Lie groups can be viewed as their direct limit, with applications to diffeomorphism groups and test function groups.

## Contribution

It proves that regular Lie groups with a direct limit chart are the direct limit of their subgroups, extending the understanding of infinite-dimensional Lie groups.

## Key findings

- Regular Lie groups with a direct limit chart are the direct limit of their subgroups.
- The group of compactly supported diffeomorphisms is a direct limit of groups supported in compact subsets.
- Similar results hold for test function groups with values in a Lie group.

## Abstract

Let G be a regular Lie group which is a directed union of regular Lie groups G_i (all modelled on possibly infinite-dimensional, locally convex spaces). We show that G is the direct limit of the G_i as a regular Lie group whenever G admits a so-called direct limit chart. Notably, this allows the regular Lie group Diff_c(M) of compactly supported smooth diffeomorphisms to be interpreted as a direct limit of the regular Lie groups Diff_K(M) of smooth diffeomorphisms supported in compact subsets K of M, even if the finite-dimensional smooth manifold M is merely paracompact (but not necessarily sigma-compact), which was not known before. Similar results are obtained for the test function groups C^k_c(M,F) with values in a Lie group F.