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Multisymplectic Geometry and Lie Groupoids

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Geometry, Mechanics, and Dynamics

Part of the book series: Fields Institute Communications ((FIC,volume 73))

Abstract

We study higher-degree generalizations of symplectic groupoids, referred to as multisymplectic groupoids. Recalling that Poisson structures may be viewed as infinitesimal counterparts of symplectic groupoids, we describe “higher” versions of Poisson structures by identifying the infinitesimal counterparts of multisymplectic groupoids. Some basic examples and features are discussed.

In memory of Jerry Marsden

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Notes

  1. 1.

    For example, in the description of the interplay between hamiltonian dynamics and symmetries [29], and in the transition from classical to quantum mechanics [7].

  2. 2.

    Here the fibred product \(\mathcal{G}_{\mathsf{s}} \times _{\mathsf{t}}\mathcal{G} =\{ (g,h) \in \mathcal{G}\times \mathcal{G}\,\vert \,\mathsf{s}(g) = \mathsf{t}(h)\}\) represents the space of composable arrows.

  3. 3.

    For a function \(f \in \varOmega ^{0}(\mathcal{G}) = C^{\infty }(\mathcal{G})\), condition (5) becomes \(f(gh) = f(g) + f(h)\), i.e., it says that f is a groupoid morphism into \(\mathbb{R}\) (viewed as an abelian group).

  4. 4.

    See e.g. [37] for a nonintegrable example and [14] for a discussion of obstructions to integrability.

  5. 5.

    In the case of exact k-plectic manifolds, a different way to eliminate the jacobiator defect is presented in [16], based on a modification of the bracket (11) using the k-plectic potential.

  6. 6.

    We say that two IM (k + 1)-forms \(\mu _{1}: A_{1} \rightarrow \wedge ^{k}T^{{\ast}}M\) and \(\mu _{2}: A_{2} \rightarrow \wedge ^{k}T^{{\ast}}M\) are equivalent if there is a Lie-algebroid isomorphism ϕ: A 1 → A 2 such that μ 2 ∘ϕ = μ 1; these are infinitesimal versions of isomorphism of Lie groupoids preserving multiplicative forms.

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Acknowledgements

H.B. and A.C. thank the organizers of the Focus Program on Geometry, Mechanics and Dynamics: the Legacy of Jerry Marsden, held at the Fields Institute in July of 2012, for their hospitality during the program, as well as MITACS for travel support (for which we also thank J. Koiller). H. B. and A. C. were partly funded by CNPq, Faperj and Capes (through the grant PVE 11/2012) D. I. thanks MICINN (Spain) for a “Ramón y Cajal” research contract; he is partially supported by MICINN grants MTM2012-34478 and MTM2009-08166-E and Canary Islands government project SOLSUB200801000238. We have benefited from many stimulating conversations with M. Forger, J.C. Marrero, N. Martinez, C. Rogers and M. Zambon. We also thank the referees for several useful comments that improved the presentation of this note.

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Authors and Affiliations

  1. IMPA, Estrada Dona Castorina 110, 22460-320, Rio de Janeiro, Brazil

    Henrique Bursztyn

  2. Departamento de Matematica Aplicada, Instituto de Matematica, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, Rio de Janeiro, RJ, 21941-909, Brasil

    Alejandro Cabrera

  3. Departamento de Matemáticas, Estadística e Investigación Operativa, Universidad de la Laguna, Santa Cruz de Tenerife, Spain

    David Iglesias

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  1. Henrique Bursztyn
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  2. Alejandro Cabrera
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Correspondence to Henrique Bursztyn .

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Editors and Affiliations

  1. Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada

    Dong Eui Chang

  2. Department of Mathematics, Imperial College London, London, United Kingdom

    Darryl D. Holm

  3. Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Saskatchewan, Canada

    George Patrick

  4. Section de Mathématiques, Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland

    Tudor Ratiu

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Bursztyn, H., Cabrera, A., Iglesias, D. (2015). Multisymplectic Geometry and Lie Groupoids. In: Chang, D., Holm, D., Patrick, G., Ratiu, T. (eds) Geometry, Mechanics, and Dynamics. Fields Institute Communications, vol 73. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2441-7_3

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