September  2021, 13(3): 517-532. doi: 10.3934/jgm.2021016

Holonomy transformations for Lie subalgebroids

KU Leuven, Department of Mathematics, Celestijnenlaan 200B Box 2400, BE-3001 Leuven, Belgium

* Corresponding author: Marco Zambon

Received  March 2021 Revised  July 2021 Published  September 2021 Early access  August 2021

Fund Project: We acknowledge partial support by the long term structural funding - Methusalem grant of the Flemish Government, the FWO and FNRS under EOS project G0H4518N, the FWO research project G083118N (Belgium)

Given a foliation, there is a well-known notion of holonomy, which can be understood as an action that differentiates to the Bott connection on the normal bundle. We present an analogous notion for Lie subalgebroids, consisting of an effective action of the minimal integration of the Lie subalgebroid, and provide an explicit description in terms of conjugation by bisections. The construction is done in such a way that it easily extends to singular subalgebroids, which provide our main motivation.

Citation: Marco Zambon. Holonomy transformations for Lie subalgebroids. Journal of Geometric Mechanics, 2021, 13 (3) : 517-532. doi: 10.3934/jgm.2021016
References:
[1]

I. Androulidakis and G. Skandalis, The holonomy groupoid of a singular foliation, J. Reine Angew. Math., 626 (2009), 1-37.  doi: 10.1515/CRELLE.2009.001.

[2]

I. Androulidakis and M. Zambon, Holonomy transformations for singular foliations, Adv. Math., 256 (2014), 348-397.  doi: 10.1016/j.aim.2014.02.003.

[3]

I. Androulidakis and M. Zambon, Integration of singular subalgebroids, preprint, arXiv: 2008.07976v1.

[4]

Z. ChenM. Stiénon and P. Xu, From Atiyah classes to homotopy Leibniz algebras, Comm. Math. Phys., 341 (2016), 309-349.  doi: 10.1007/s00220-015-2494-6.

[5]

M. Crainic, Differentiable and algebroid cohomology, van Est isomorphisms, and characteristic classes, Comment. Math. Helv., 78 (2003), 681-721.  doi: 10.1007/s00014-001-0766-9.

[6]

A. Garmendia and J. Villatoro, Integration of singular foliations via paths, preprint, arXiv: 1912.02148, to appear in International Mathematics Research Notices.

[7]

J. L. Heitsch, A cohomology for foliated manifolds, Bull. Amer. Math. Soc., 79 (1973), 1283–1285.. doi: 10.1090/S0002-9904-1973-13416-0.

[8]

C. Laurent-Gengoux and Y. Voglaire, Invariant connections and PBW theorem for Lie groupoid pairs, Pacific J. Math., 303 (2019), 605-667.  doi: 10.2140/pjm.2019.303.605.

[9] K. C. H. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, London Mathematical Society Lecture Note Series, 213, Cambridge University Press, Cambridge, 2005.  doi: 10.1017/CBO9781107325883.
[10] I. Moerdijk and J. Mrčun, Introduction to Foliations and Lie Groupoids, Cambridge Studies in Advanced Mathematics, 91, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511615450.
[11]

I. Moerdijk and J. Mrčun, On the integrability of Lie subalgebroids, Adv. Math., 204 (2006), 101-115.  doi: 10.1016/j.aim.2005.05.011.

[12]

M. Zambon, Singular subalgebroids, preprint, arXiv: 1805.02480, to appear in Annales de l’Institut Fourier. Appendix by I. Androulidakis.

show all references

References:
[1]

I. Androulidakis and G. Skandalis, The holonomy groupoid of a singular foliation, J. Reine Angew. Math., 626 (2009), 1-37.  doi: 10.1515/CRELLE.2009.001.

[2]

I. Androulidakis and M. Zambon, Holonomy transformations for singular foliations, Adv. Math., 256 (2014), 348-397.  doi: 10.1016/j.aim.2014.02.003.

[3]

I. Androulidakis and M. Zambon, Integration of singular subalgebroids, preprint, arXiv: 2008.07976v1.

[4]

Z. ChenM. Stiénon and P. Xu, From Atiyah classes to homotopy Leibniz algebras, Comm. Math. Phys., 341 (2016), 309-349.  doi: 10.1007/s00220-015-2494-6.

[5]

M. Crainic, Differentiable and algebroid cohomology, van Est isomorphisms, and characteristic classes, Comment. Math. Helv., 78 (2003), 681-721.  doi: 10.1007/s00014-001-0766-9.

[6]

A. Garmendia and J. Villatoro, Integration of singular foliations via paths, preprint, arXiv: 1912.02148, to appear in International Mathematics Research Notices.

[7]

J. L. Heitsch, A cohomology for foliated manifolds, Bull. Amer. Math. Soc., 79 (1973), 1283–1285.. doi: 10.1090/S0002-9904-1973-13416-0.

[8]

C. Laurent-Gengoux and Y. Voglaire, Invariant connections and PBW theorem for Lie groupoid pairs, Pacific J. Math., 303 (2019), 605-667.  doi: 10.2140/pjm.2019.303.605.

[9] K. C. H. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, London Mathematical Society Lecture Note Series, 213, Cambridge University Press, Cambridge, 2005.  doi: 10.1017/CBO9781107325883.
[10] I. Moerdijk and J. Mrčun, Introduction to Foliations and Lie Groupoids, Cambridge Studies in Advanced Mathematics, 91, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511615450.
[11]

I. Moerdijk and J. Mrčun, On the integrability of Lie subalgebroids, Adv. Math., 204 (2006), 101-115.  doi: 10.1016/j.aim.2005.05.011.

[12]

M. Zambon, Singular subalgebroids, preprint, arXiv: 1805.02480, to appear in Annales de l’Institut Fourier. Appendix by I. Androulidakis.

Figure 1.  The Lie groupoid $ G $, with the right-invariant distribution $ \overset{\rightarrow }{{B}} $ (in blue). An element $ \xi\in H(\overset{\rightarrow }{B}) $ induces a diffeomorphism $ \Xi(\xi) $ between slices (slices not depicted). It is related to $ \Xi(\xi\cdot g) $ as explained in Step 1 of Theorem 2.1
Figure 2.  Given $ h\in H^G(B) $, the construction of the map $ \chi(h) $ as in Step 2 of Theorem 2.1. In this picture the slices $ S $ are zero-dimensional, but are depicted as short red segments
Figure 3.  Given $ h\in H^G(B) $, the construction of the map $ \chi^{conj}(h) $. The slices $ S $ are zero-dimensional, but are depicted as short red segments
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