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.
Revised:
Accepted:
Published online:
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
@article{AIF_2022__72_6_2109_0, author = {Zambon, Marco}, title = {Singular subalgebroids}, journal = {Annales de l'Institut Fourier}, pages = {2109--2190}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {72}, number = {6}, year = {2022}, doi = {10.5802/aif.3493}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3493/} }
TY - JOUR AU - Zambon, Marco TI - Singular subalgebroids JO - Annales de l'Institut Fourier PY - 2022 SP - 2109 EP - 2190 VL - 72 IS - 6 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3493/ DO - 10.5802/aif.3493 LA - en ID - AIF_2022__72_6_2109_0 ER -
Zambon, Marco. Singular subalgebroids. Annales de l'Institut Fourier, Volume 72 (2022) no. 6, pp. 2109-2190. doi : 10.5802/aif.3493. https://aif.centre-mersenne.org/articles/10.5802/aif.3493/
[1] The holonomy groupoid of a singular foliation, J. Reine Angew. Math., Volume 626 (2009), pp. 1-37 | DOI | MR | Zbl
[2] The analytic index of elliptic pseudodifferential operators on a singular foliation, J. K-Theory, Volume 8 (2011) no. 3, pp. 363-385 | DOI | MR | Zbl
[3] Pseudodifferential calculus on a singular foliation, J. Noncommut. Geom., Volume 5 (2011) no. 1, pp. 125-152 | DOI | MR | Zbl
[4] Smoothness of holonomy covers for singular foliations and essential isotropy, Math. Z., Volume 275 (2013) no. 3-4, pp. 921-951 | DOI | MR | Zbl
[5] Holonomy transformations for singular foliations, Adv. Math., Volume 256 (2014), pp. 348-397 | DOI | MR | Zbl
[6] Almost regular Poisson manifolds and their holonomy groupoids, Selecta Math. (N.S.), Volume 23 (2017) no. 3, pp. 2291-2330 | DOI | MR | Zbl
[7] Integration of Singular Subalgebroids (2020) (https://arxiv.org/abs/2008.07976)
[8] Geometric models for noncommutative algebras, Berkeley Mathematics Lecture Notes, 10, American Mathematical Society, Providence, RI, 1999, xiv+184 pages | MR | Zbl
[9] Lectures on integrability of Lie brackets, Lectures on Poisson geometry (Geom. Topol. Monogr.), Volume 17, Geom. Topol. Publ., Coventry, 2011, pp. 1-107 | DOI | MR
[10] Hausdorff Morita equivalence of singular foliations, Ann. Global Anal. Geom., Volume 55 (2019) no. 1, pp. 99-132 | DOI | MR | Zbl
[11] Quotients of singular foliations and Lie 2-group actions, J. Noncommut. Geom., Volume 15 (2021) no. 4, pp. 1251-1283 | DOI | MR | Zbl
[12] Symplectic groupoids of log symplectic manifolds, Int. Math. Res. Not. IMRN (2014) no. 11, pp. 3022-3074 | DOI | MR | Zbl
[13] Lie groupoids and their orbispaces, Port. Math., Volume 70 (2013) no. 2, pp. 161-209 | DOI | MR | Zbl
[14] Poisson–Nijenhuis structures, Ann. Inst. H. Poincaré Phys. Théor., Volume 53 (1990) no. 1, pp. 35-81 | MR | Zbl
[15] The universal Lie -algebroid of a singular foliation, Doc. Math., Volume 25 (2020), pp. 1571-1652 | DOI | MR | Zbl
[16] Lie -algebroids and singular foliations, Ph. D. Thesis, Université de Lyon (2017) (https://arxiv.org/abs/1703.07404)
[17] General theory of Lie groupoids and Lie algebroids, London Mathematical Society Lecture Note Series, 213, Cambridge University Press, Cambridge, 2005, xxxviii+501 pages | DOI | MR | Zbl
[18] On the integrability of Lie subalgebroids, Adv. Math., Volume 204 (2006) no. 1, pp. 101-115 | DOI | MR | Zbl
[19] The holonomic imperative and the homotopy groupoid of a foliated manifold, Rocky Mountain J. Math., Volume 17 (1987) no. 1, pp. 151-165 | DOI | MR | Zbl
[20] Differential forms on general commutative algebras, Trans. Amer. Math. Soc., Volume 108 (1963), pp. 195-222 | DOI | MR | Zbl
Cited by Sources: