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Title: Lie groupoids of mappings taking values in a Lie groupoid (English)
Author: Amiri, Habib
Author: Glöckner, Helge
Author: Schmeding, Alexander
Language: English
Journal: Archivum Mathematicum
ISSN: 0044-8753 (print)
ISSN: 1212-5059 (online)
Volume: 56
Issue: 5
Year: 2020
Pages: 307-356
Summary lang: English
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Category: math
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Summary: Endowing differentiable functions from a compact manifold to a Lie group with the pointwise group operations one obtains the so-called current groups and, as a special case, loop groups. These are prime examples of infinite-dimensional Lie groups modelled on locally convex spaces. In the present paper, we generalise this construction and show that differentiable mappings on a compact manifold (possibly with boundary) with values in a Lie groupoid form infinite-dimensional Lie groupoids which we call current groupoids. We then study basic differential geometry and Lie theory for these Lie groupoids of mappings. In particular, we show that certain Lie groupoid properties, like being a proper étale Lie groupoid, are inherited by the current groupoid. Furthermore, we identify the Lie algebroid of a current groupoid as a current algebroid (analogous to the current Lie algebra associated to a current Lie group). To establish these results, we study superposition operators \[ C^\ell (K,f)\colon C^\ell (K,M)\rightarrow C^\ell (K,N)\,,\;\, \gamma f\circ \gamma \] between manifolds of $C^\ell $-functions. Under natural hypotheses, $C^\ell (K,f)$ turns out to be a submersion (an immersion, an embedding, proper, resp., a local diffeomorphism) if so is the underlying map $f\colon M\rightarrow N$. These results are new in their generality and of independent interest. (English)
Keyword: Lie groupoid
Keyword: Lie algebroid
Keyword: topological groupoid
Keyword: mapping groupoid
Keyword: current groupoid
Keyword: manifold of mappings
Keyword: superposition operator
Keyword: Nemytskii operator
Keyword: pushforward
Keyword: submersion
Keyword: immersion
Keyword: embedding
Keyword: local diffeomorphism
Keyword: \mathbbT1-cmd 1étale map
Keyword: proper map
Keyword: perfect map
Keyword: orbifold groupoid
Keyword: transitivity
Keyword: local transitivity
Keyword: local triviality
Keyword: Stacey-Roberts Lemma
MSC: 22A22
MSC: 22E65
MSC: 22E67
MSC: 46T10
MSC: 47H30
MSC: 58D15
MSC: 58H05
idZBL: Zbl 07285968
idMR: MR4188745
DOI: 10.5817/AM2020-5-307
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Date available: 2020-11-20T14:00:58Z
Last updated: 2021-02-23
Stable URL: http://hdl.handle.net/10338.dmlcz/148441
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