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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMay 28th 2015
   • (edited May 28th 2015)
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded to _[[Poisson manifold]]_ a subsection * _[Presymplectic manifolds and infinitesimal quantomorphisms](http://ncatlab.org/nlab/show/Poisson+manifold#PresymplecticManifolds)_ 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.

   added to Poisson manifold a subsection

   • Presymplectic manifolds and infinitesimal quantomorphisms

   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.

2. • CommentRowNumber2.
   • CommentAuthorDmitri Pavlov
   • CommentTimeMar 20th 2023
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   Author: Dmitri Pavlov
   Format: MarkdownItexAdded: 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](http://dx.doi.org/10.4310/jdg/1214433987). <a href="https://ncatlab.org/nlab/revision/diff/Poisson+manifold/36">diff</a>, <a href="https://ncatlab.org/nlab/revision/Poisson+manifold/36">v36</a>, <a href="https://ncatlab.org/nlab/show/Poisson+manifold">current</a>

   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

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeDec 4th 2023
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to: * [[Alan Weinstein]], *The local structure of Poisson manifolds*, J. Differential Geom. **18** 3 (1983) 523-557 &lbrack;[doi:10.4310/jdg/1214437787](https://projecteuclid.org/journals/journal-of-differential-geometry/volume-18/issue-3/The-local-structure-of-Poisson-manifolds/10.4310/jdg/1214437787.full)&rbrack; <a href="https://ncatlab.org/nlab/revision/diff/Poisson+manifold/39">diff</a>, <a href="https://ncatlab.org/nlab/revision/Poisson+manifold/39">v39</a>, <a href="https://ncatlab.org/nlab/show/Poisson+manifold">current</a>

   added pointer to:

   • Alan Weinstein, The local structure of Poisson manifolds, J. Differential Geom. 18 3 (1983) 523-557 [doi:10.4310/jdg/1214437787]

   diff, v39, current

4. • CommentRowNumber4.
   • CommentAuthorperezl.alonso
   • CommentTimeJun 23rd 2025
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   Author: perezl.alonso
   Format: MarkdownItexIs there a smooth moduli stack of Poisson structures?

   Is there a smooth moduli stack of Poisson structures?

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeJun 23rd 2025
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   Author: Urs
   Format: MarkdownItexLet's see. Poisson structures on a smooth manifold $X$ are equivalently bivector fields $\pi$ whose [[Schouten bracket]] with themselves vanishes: $[\pi,\pi] = 0$. (This ought to be prominently explained on the nLab, but maybe it isn't. We have a remark at *[[Schouten bracket]]* [here](https://ncatlab.org/nlab/show/Schouten+bracket#PoissonStructures) and at *[[symplectic Lie n-algebroid]]* [here](https://ncatlab.org/nlab/show/symplectic+Lie+n-algebroid#PoissonManifolds).) By shifting the naive degree down by one, the Schouten bracket makes the multivector fields $\wedge^\bullet T X$ into a graded Lie algebra, hence into a dg-Lie algebra whose differential happens to vanish, and so the Poisson structures $\pi$ 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]].

   Let’s see.

   Poisson structures on a smooth manifold are equivalently bivector fields whose Schouten bracket with themselves vanishes: .

   (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 into a graded Lie algebra, hence into a dg-Lie algebra whose differential happens to vanish, and so the Poisson structures 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.

6. • CommentRowNumber6.
   • CommentAuthorperezl.alonso
   • CommentTimeJun 23rd 2025
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   Author: perezl.alonso
   Format: MarkdownItexHmm, I see, thanks, Urs.

   Hmm, I see, thanks, Urs.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeJun 23rd 2025
   • PermaLink
   Author: Urs
   Format: MarkdownItexThis would be a great example to spell out a little more on the nLab!

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