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Poisson and Symplectic Functions in Lie Algebroid Theory

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Higher Structures in Geometry and Physics

Part of the book series: Progress in Mathematics ((PM,volume 287))

Abstract

Emphasizing the role of Gerstenhaber algebras and of higher derived brackets in the theory of Lie algebroids, we show that the several Lie algebroid brackets which have been introduced in the recent literature can all be defined in terms of Poisson and pre-symplectic functions in the sense of Roytenberg and Terashima. We prove that in this very general framework there exists a one-to-one correspondence between nondegenerate Poisson functions and symplectic functions. We also determine the differential associated to a Lie algebroid structure obtained by twisting a structure with background by both a Lie bialgebra action and a Poisson bivector.

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Notes

  1. 1.

    In [23, 5], Lie-quasi bialgebras were called Jacobian quasi-bialgebras, and quasi-Lie bialgebras were called co-Jacobian quasi-bialgebras. We also point out that, in the translation of Drinfeld’s original paper [13], the term “quasi-Lie bialgebra” is used for what we call Lie-quasi bialgebra. Proto-bialgebras were introduced in [23] where they were called proto-Lie-bialgebras, to distinguish them from the associative version of this notion.

  2. 2.

    There are some changes in the notations. In particular, the notations ϕ and ψ used by Roytenberg in [46] are exchanged in order to return to the conventions of [23, 5, 27].

  3. 3.

    Even Poisson brackets had already appeared in the context of the quantization of systems with constraints in the work of Batalin, Fradkin and Vilkovisky. See [50] and references therein.

  4. 4.

    The Koszul bracket [33] restricts to the bracket of sections of Γ(V ∗) generalizing the well-known bracket of 1-forms on a Poisson manifold. The bracket of 1-forms on symplectic manifolds was introduced in the book of Abraham and Marsden (1967). For Poisson manifolds, it was discovered independently in the 1980s by several authors – Gelfand and Dorfman, Fuchssteiner, Magri and Morosi, Daletskii – and Weinstein (see [9]) has shown that it is a Lie algebroid bracket.

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Acknowledgements

The main results of this paper were presented at the international conference “Higher Structures in Geometry and Physics” which was held in honor of Murray Gerstenhaber and Jim Stasheff at the Institut Henri Poincaré in Paris in January 2007. I am very grateful to the organizers, Alberto Cattaneo and Ping Xu, for the invitation to participate in this exciting conference.I thank Murray Gerstenhaber, Jim Stasheff, and Dmitry Roytenberg for their remarks and useful conversations.

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  1. Centre de Mathématiques Laurent Schwartz, École Polytechnique, 91128, Palaiseau, France

    Yvette Kosmann-Schwarzbach

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  1. Yvette Kosmann-Schwarzbach
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Correspondence to Yvette Kosmann-Schwarzbach .

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  1. Universität Zürich, Institut für Mathematik, Winterthurerstr. 190, Zürich, CH-8057, Switzerland

    Alberto S. Cattaneo

  2. Loyola University, Chicago, Department of Mathematics, Chicago, 60626, Illinois, USA

    Anthony Giaquinto

  3. Pennsylvania State University, Department of Mathematics, Eberly College of Science, University Park, 16802, Pennsylvania, USA

    Ping Xu

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Kosmann-Schwarzbach, Y. (2011). Poisson and Symplectic Functions in Lie Algebroid Theory. In: Cattaneo, A., Giaquinto, A., Xu, P. (eds) Higher Structures in Geometry and Physics. Progress in Mathematics, vol 287. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-4735-3_12

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