Linking Lie groupoid representations and representations of infinite-dimensional Lie groups. (English) Zbl 1416.22022
In this paper, the authors show that smooth representations of Lie groupoids produce representations of associated Lie groups. These groups are the bisection group and a group of groupoid self-maps.
So, representations of the Lie groupoids produce 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 authors obtain indeed continuous and smooth representations.
In the topological category, this correspondence can be reversed for certain topological groupoids. The authors extend this result to the smooth category under weaker assumptions on the groupoids.
So, representations of the Lie groupoids produce 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 authors obtain indeed continuous and smooth representations.
In the topological category, this correspondence can be reversed for certain topological groupoids. The authors extend this result to the smooth category under weaker assumptions on the groupoids.
Reviewer: Marta Macho Stadler (Leioa)
MSC:
| 22E66 | Analysis on and representations of infinite-dimensional Lie groups |
| 22E65 | Infinite-dimensional Lie groups and their Lie algebras: general properties |
| 22A22 | Topological groupoids (including differentiable and Lie groupoids) |
| 58D15 | Manifolds of mappings |
Keywords:
Lie groupoid; representation of groupoids; group of bisections; infinite-dimensional Lie group; smooth representation; semi-linear map; jet groupoidReferences:
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