Lie groupoids of mappings taking values in a Lie groupoid

Habib Amiri; Helge Glöckner; Alexander Schmeding

Archivum Mathematicum (2020)

  • Volume: 056, Issue: 5, page 307-356
  • ISSN: 0044-8753

Abstract

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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 ℓ ( K , f ) : C ℓ ( K , M ) → C ℓ ( K , N ) , γ f ∘ γ between manifolds of C ℓ -functions. Under natural hypotheses, C ℓ ( K , f ) turns out to be a submersion (an immersion, an embedding, proper, resp., a local diffeomorphism) if so is the underlying map f : M → N . These results are new in their generality and of independent interest.

How to cite

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Amiri, Habib, Glöckner, Helge, and Schmeding, Alexander. "Lie groupoids of mappings taking values in a Lie groupoid." Archivum Mathematicum 056.5 (2020): 307-356. <http://eudml.org/doc/297032>.

@article{Amiri2020,
abstract = {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.},
author = {Amiri, Habib, Glöckner, Helge, Schmeding, Alexander},
journal = {Archivum Mathematicum},
keywords = {Lie groupoid; Lie algebroid; topological groupoid; mapping groupoid; current groupoid; manifold of mappings; superposition operator; Nemytskii operator; pushforward; submersion; immersion; embedding; local diffeomorphism; 1-cmd 1étale map; proper map; perfect map; orbifold groupoid; transitivity; local transitivity; local triviality; Stacey-Roberts Lemma},
language = {eng},
number = {5},
pages = {307-356},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Lie groupoids of mappings taking values in a Lie groupoid},
url = {http://eudml.org/doc/297032},
volume = {056},
year = {2020},
}

TY - JOUR
AU - Amiri, Habib
AU - Glöckner, Helge
AU - Schmeding, Alexander
TI - Lie groupoids of mappings taking values in a Lie groupoid
JO - Archivum Mathematicum
PY - 2020
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 056
IS - 5
SP - 307
EP - 356
AB - 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.
LA - eng
KW - Lie groupoid; Lie algebroid; topological groupoid; mapping groupoid; current groupoid; manifold of mappings; superposition operator; Nemytskii operator; pushforward; submersion; immersion; embedding; local diffeomorphism; 1-cmd 1étale map; proper map; perfect map; orbifold groupoid; transitivity; local transitivity; local triviality; Stacey-Roberts Lemma
UR - http://eudml.org/doc/297032
ER -

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