# Linking Lie groupoid representations and representations of infinite-dimensional Lie groups

## Abstract

The present paper links the representation theory of Lie groupoids and infinite-dimensional Lie groups. We show that smooth representations of Lie groupoids give rise to smooth representations of associated Lie groups. The groups envisaged here are the bisection group and a group of groupoid self-maps. Then, representations of the Lie groupoids give rise to representations of the infinite-dimensional Lie groups on spaces of (compactly supported) bundle sections. Endowing the spaces of bundle sections with a fine Whitney type topology, the fine very strong topology, we even obtain continuous and smooth representations. It is known that in the topological category, this correspondence can be reversed for certain topological groupoids. We extend this result to the smooth category under weaker assumptions on the groupoids.

## Keywords

Lie groupoid Representation of groupoids Group of bisections Infinite-dimensional Lie group Smooth representation Semi-linear map Jet groupoid## Mathematics Subject Classification

Primary: 22E66 Secondary: 22E65 22A22 58D15## Notes

### Acknowledgements

The authors thank K.–H. Neeb for helpful conversations on the subject of this work. We also thank the anonymous referee for numerous comments and suggestions which helped improve the article.

## Supplementary material

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