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Lie rackoids

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Abstract

We define a new differential geometric structure, called Lie rackoid. A Lie rackoid differentiates to Leibniz algebroids in a similar way as Lie groupoids differentiate to Lie algebroids. Its main ingredient is a self-distributive product on the manifold of bisections of a smooth precategory. We show that the tangent algebroid of a Lie rackoid is a Leibniz algebroid and that Lie groupoids give rise via conjugation to a Lie rackoid. Our main objective are large classes of examples, including a Lie rackoid integrating the Dorfman bracket without the cocycle term of the standard Courant algebroid.

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Notes

  1. The concept of a Leibniz algebra appears to our knowledge first in a paper by Blokh in 1965.

  2. Equivalently, for all bisections \(\Sigma \) and all elements \(\gamma \in \Gamma \), the composition \( \sigma \rhd \gamma \) is defined to be an element of \(\Gamma \) whose source is \(\underline{\sigma } \circ s (\gamma ) \) and whose target is \(\underline{\sigma } \circ t (\gamma ) \).

  3. Note that this condition implies that the p-image of a bisection of X is a bisection of \(\Gamma \).

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Acknowledgments

FW thanks Université de Lorraine for financing several visits to Metz where this research has been carried out. C. Laurent-Gengoux and F. Wagemann thank Christoph Wockel for useful discussions. They also thank the anonymous referee for suggestions which led to numerous improvements of the manuscript.

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Authors and Affiliations

  1. Institut Elie Cartan de Lorraine, Université de Lorraine, Île du Saulcy, 57000, Metz, France

    Camille Laurent-Gengoux

  2. Laboratoire de mathématiques Jean Leray, 2, rue de la Houssinière, 44322, Nantes cedex 3, France

    Friedrich Wagemann

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  1. Camille Laurent-Gengoux
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  2. Friedrich Wagemann
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Correspondence to Friedrich Wagemann.

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Laurent-Gengoux, C., Wagemann, F. Lie rackoids. Ann Glob Anal Geom 50, 187–207 (2016). https://doi.org/10.1007/s10455-016-9507-3

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  • DOI: https://doi.org/10.1007/s10455-016-9507-3

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