Smooth Lie group actions are parametrized diffeological subgroups

TL;DR

The paper demonstrates that effective smooth Lie group actions on manifolds can be viewed as parametrized diffeological subgroups of the diffeomorphism group, with the action map being a diffeomorphism onto its image.

Contribution

It establishes that such Lie group actions are precisely diffeological subgroups of Diff(M), providing a new perspective on their structure.

Findings

Effective smooth Lie group actions are diffeomorphisms onto their images.

Images of these actions form diffeological subgroups of Diff(M).

The action map is a diffeomorphism with respect to the subset diffeology.

Abstract

We show that every effective smooth action of a Lie group G on a manifold M is a diffeomorphism from G onto its image in Diff(M), where the image is equipped with the subset diffeology of the functional diffeology.