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Glanon groupoids

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Abstract

We introduce the notions of Glanon groupoids, which are Lie groupoids equipped with multiplicative generalized complex structures, and of Glanon algebroids, their infinitesimal counterparts. Both symplectic and holomorphic Lie groupoids are particular instances of Glanon groupoids. We prove that there is a bijection between Glanon algebroids on one hand and source connected and source-simply connected Glanon groupoids on the other. As a consequence, we recover various known integrability results and obtain the integration of holomorphic Lie bialgebroids to holomorphic Poisson groupoids.

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Notes

  1. Yoshimura and Marsden refer to the Whitney sum \(TM\oplus T^*M\) as the “Pontryagin bundle” of \(M\) because of the fundamental role it plays in the geometric interpretation of Pontryagin’s maximum principle. Izu Vaisman calls it the “big tangent bundle” of \(M\). For more details, see [22].

  2. The Whitney sum \(T\varGamma \oplus T^*\varGamma \), which is simultaneously a vector bundle with \(\varGamma \) as base manifold and a Lie groupoid with \(TM\oplus A^*\) as unit space, is a VB-groupoid with the vector bundle \(A\oplus T^*M\) over \(M\) as core [26, 32]. It is well known that a morphism of VB-groupoids determines a morphism of their cores.

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Acknowledgments

We would like to thank Camille Laurent-Gengoux, Rajan Mehta, and Cristian Ortiz for useful discussions and an anonymous referee for suggesting many improvements. We would also like to thank several institutions for their hospitality while work on this project was underway: Penn State (Jotz Lean), and Institut des Hautes Études Scientifiques and Beijing International Center for Mathematical Research (Xu). Special thanks go to the Michéa family and our many friends in Glanon, whose warmth provided us constant inspiration for our work in the past. Our memories of our many friends from Glanon who have unfortunately left mathematics serve as a powerful reminder that our primary role as mathematicians is to elucidate mathematics while enjoying its beauty. We dedicate this paper to our dear old friends.

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Authors and Affiliations

  1. School of Mathematics and Statistics, The University of Sheffield, Sheffield, UK

    Madeleine Jotz Lean

  2. Department of Mathematics, Penn State University, State College, USA

    Mathieu Stiénon & Ping Xu

Authors
  1. Madeleine Jotz Lean
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  2. Mathieu Stiénon
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  3. Ping Xu
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Correspondence to Madeleine Jotz Lean.

Additional information

Research partially supported by a Dorothea-Schlözer fellowship of the University of Göttingen, Swiss NSF Grants 200021-121512 and PBELP2_137534, NSF Grants DMS-0801129 and DMS-1101827, NSA Grant H98230-12-1-0234.

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Lean, M.J., Stiénon, M. & Xu, P. Glanon groupoids. Math. Ann. 364, 485–518 (2016). https://doi.org/10.1007/s00208-015-1222-z

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  • DOI: https://doi.org/10.1007/s00208-015-1222-z

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