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Holonomy transformations for Lie subalgebroids

  • * Corresponding author: Marco Zambon

    * Corresponding author: Marco Zambon

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)

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  • 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.

    Mathematics Subject Classification: Primary: 22A22, 53C29; Secondary: 53C12, 53D17.


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  • 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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    [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.
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    [9] K. C. H. MackenzieGeneral Theory of Lie Groupoids and Lie Algebroids, London Mathematical Society Lecture Note Series, 213, Cambridge University Press, Cambridge, 2005.  doi: 10.1017/CBO9781107325883.
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