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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMay 28th 2015
    • (edited May 28th 2015)

    added to Poisson manifold a subsection

    which states the form of the Poisson bracket on presymplectic manifolds and then discusses how this is isomorphic as a Lie algebra to the infinitesimal quantomorphisms, the infinitesimal symmetries of any prequantization of the presymplectic manifold.

    I have written that as one subsection for the exposition at geometry of physics – prequantum geometry, but I suppose it serves well to have it right there in the entry on Poisson brackets itsef, too.

    • CommentRowNumber2.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMar 20th 2023

    Added:

    The original reference is

    • André Lichnerowicz, Les variétés de Poisson et leurs algèbres de Lie associées, Journal of Differential Geometry 12:2 (1977), 253–300. doi.

    diff, v36, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeDec 4th 2023

    added pointer to:

    diff, v39, current

    • CommentRowNumber4.
    • CommentAuthorperezl.alonso
    • CommentTimeJun 23rd 2025

    Is there a smooth moduli stack of Poisson structures?

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJun 23rd 2025

    Let’s see.

    Poisson structures on a smooth manifold X are equivalently bivector fields π whose Schouten bracket with themselves vanishes: [π,π]=0.

    (This ought to be prominently explained on the nLab, but maybe it isn’t. We have a remark at Schouten bracket here and at symplectic Lie n-algebroid here.)

    By shifting the naive degree down by one, the Schouten bracket makes the multivector fields TX into a graded Lie algebra, hence into a dg-Lie algebra whose differential happens to vanish, and so the Poisson structures π are equivalently the Maurer-Cartan elements in this dg-Lie algebra.

    From this one gets the derived moduli stack of Poisson structures by the machinery of formal moduli problems.

    • CommentRowNumber6.
    • CommentAuthorperezl.alonso
    • CommentTimeJun 23rd 2025

    Hmm, I see, thanks, Urs.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJun 23rd 2025

    This would be a great example to spell out a little more on the nLab!