Holonomy transformations for Lie subalgebroids

Zambon Marco

doi:10.3934/jgm.2021016

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2021, Volume 13, Issue 3: 517-532. Doi: 10.3934/jgm.2021016

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Holonomy transformations for Lie subalgebroids

Zambon Marco^,

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

^* Corresponding author: Marco Zambon
^* Corresponding author: Marco Zambon

Received: February 28, 2021

Revised: June 30, 2021

Early access: August 2021

Published: September 2021

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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Abstract

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.

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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

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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

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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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References

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