Singular subalgebroids
Annales de l'Institut Fourier, Volume 72 (2022) no. 6, pp. 2109-2190.

We introduce singular subalgebroids of an integrable Lie algebroid, extending the notion of Lie subalgebroid by dropping the constant rank requirement. We lay the bases of a Lie theory for singular subalgebroids: we construct the associated holonomy groupoids, adapting the procedure of Androulidakis–Skandalis for singular foliations, in a way that keeps track of the choice of Lie groupoid integrating the ambient Lie algebroid. In the regular case, this recovers the integration of Lie subalgebroids by Moerdijk–Mrčun. The holonomy groupoids are topological groupoids, and are suitable for noncommutative geometry as they allow for the construction of the associated convolution algebras. Further we carry out the construction for morphisms in a functorial way.

Nous introduisons les sous-algébroïdes singuliers d’un algébroïde de Lie intégrable, qui étendent la notion de sous-algébroïde de Lie par la suppression de la condition de rang constant. Nous posons les bases d’une théorie de Lie pour les sous-algébroïdes singuliers  : nous construisons les groupoïdes d’holonomie associés, en adaptant la procédure d’Androulidakis–Skandalis pour les feuilletages singuliers, de manière à tenir compte du choix du groupoïde de Lie intégrant l’algébroïde de Lie. Dans le cas régulier, on retrouve bien l’intégration des sous-algébroïdes de Lie par Moerdijk–Mrčun. Les groupoïdes d’holonomie sont des groupoïdes topologiques, et ils sont adaptés à la géométrie non commutative car ils permettent la construction des algèbres de convolution associées. Enfin, nous réalisons la construction des morphismes de manière fonctorielle.

Published online:
DOI: 10.5802/aif.3493
Classification: 22A22, 17B66, 22E60, 53D17
Keywords: Lie subalgebroid, singular foliation, holonomy groupoid, Lie groupoid.
Mot clés : Sous-algébroïde de Lie, feuilletage singulier, groupoïde d’holonomie, groupoïde de Lie.
Zambon, Marco 1

1 KU Leuven Department of Mathematics Celestijnenlaan 200B, BE-3001 Leuven (Belgium)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Zambon, Marco. Singular subalgebroids. Annales de l'Institut Fourier, Volume 72 (2022) no. 6, pp. 2109-2190. doi : 10.5802/aif.3493.

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