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Amiri, Habib; Glöckner, Helge; Schmeding, Alexander

Lie groupoids of mappings taking values in a Lie groupoid. (English) Zbl 07285968

Arch. Math. (Brno) 56, No. 5, 307-356 (2020).
The paper under review is a very significant contribution to the incipient theory of infinite-dimensional Lie groupoids modelled on locally convex spaces. Specifically, one performs a systematic investigation of Lie groupoid structures on spaces of functions defined on a manifold and taking values in a fixed finite-dimensional Lie groupoid. In particular, one explicitly describes the Lie algebroids associated to the infinite-dimensional Lie groupoids that are constructed in this way. Remarkably, in order to achieve their aims, the authors establish some tools that clearly hold an independent interest, such as composition operators on spaces of differentiable mappings, with an emphasis on properties such as being submersion, immersion, étale, proper and so on.
Reviewer: Daniel Beltiţă (Bucureşti)

Cited in 5 Documents

MSC:

22A22 Topological groupoids (including differentiable and Lie groupoids)
22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
22E67 Loop groups and related constructions, group-theoretic treatment
46T10 Manifolds of mappings
47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)
58D15 Manifolds of mappings
58H05 Pseudogroups and differentiable groupoids

Keywords:

Lie groupoid; Lie algebroid; mapping groupoid; current groupoid; manifold of mappings; superposition operator; Nemytskii operator; local transitivity; local triviality; pushforward; submersion; immersion; embedding; local diffeomorphism; étale map; proper map; perfect map; orbifold groupoid; Stacey-Roberts lemma
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References:

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