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It was adapted from the Wikipedia page Poisson_manifold and contains some or all of that page's content licensed under a CC BY-SA license. Post-publication review comments or direct edits can be left at the version as it appears on Wikipedia.
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DOI: 10.15347/WJS/2024.006
QID: Q117054291
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Francesco Cattafi (15 July 2024). "Poisson manifold". WikiJournal of Science 7 (1): 6. doi:10.15347/WJS/2024.006. Wikidata Q117054291. ISSN 2470-6345.
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Wikipedia: This work is adapted from the Wikipedia article Poisson manifold (CC BY-SA). Content has also subsequently been used to update that same Wikipedia article Poisson manifold.
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This is an open access article distributed under the Creative Commons Attribution ShareAlike License, which permits unrestricted use, distribution, and reproduction, provided the original author and source are credited.
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Henrique Bursztyn
Eckhard Meinrenken
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Plagiarism check
Pass. Report from WMF copyvios tool only showed overlap in book titles in the reference section. OhanaUnitedTalk page 19:42, 24 June 2023 (UTC)
First peer review
Review by Henrique Bursztyn 
, IMPA
These assessment comments were submitted on , and refer to this previous version of the article
Overall I find that the article is well structured and clearly written. It highlights the main basic points of the subject and serves well the purpose of providing a quick introduction to it. I list below comments, suggestions and a few corrections.
- I find the opening sentence "In differential geometry, a field in mathematics, a Poisson manifold is a smooth manifold endowed with a Poisson structure." not illuminating. A possible suggestion is to swap the content of the first two paragraphs:
"In differential geometry, a field of mathematics, a Poisson structure (or Poisson bracket) on a smooth manifold M is....."
"A Poisson manifold is a smooth manifold endowed with a Poisson structure. The notion of Poisson manifold generalises..."
- Second paragraph of the introduction: The convention that the Poisson bracket of a symplectic form is {f,g} = \omega(X_f,X_g) is not consistently used (see e.g. the third paragraph of the section "The non-regular case"), and does not seem compatible with future claims (regarding signs).
A possible way to fix it is to write \omega= dq dp, i_{X_f}\omega=df, and {f,g}=\omega(X_g,X_f) = L_{X_f}g. This ensures that {f,.}=X_f, \pi^sharp = (\omega^\flat)^{-1}, and [X_f,X_g]=X_{f,g}
- 5th paragraph of the introduction:
the sentence "... foliation of the manifold into symplectic submanifolds" could cause confusion, it might be better to say something like:"...a foliation whose leaves are symplectic", or "...a foliation whose leaves are naturally equipped with symplectic forms"...
- 6th paragraph of the introduction: saying that "their "symplectic form" should be allowed to be degenerate" is not that accurate, since it is not the form that becomes degenerate... Perhaps "... which should be "morally" symplectic, but fail to be so. For example, ..."
- same paragraph:
"is general is" should be "in general is", "obtained *by* quotienting"
- Section "As bivector":
in the expression in display for \pi|_U, one typically puts a "1/2" *or* writes "i<j" for the sum.
- Section "Equivalence of the definitions".
The construction in the sentence "A Poisson structure without any of the four requirements above is also called an almost Poisson structure" is not ideal, since it defines almost Poisson as a Poisson manifold with (one less property) .... I think it would be better to say, before the equivalent integrability conditions, that "a bivector field, or the corresponding almost Lie bracket, is called an almost Poisson structure. An almost Poisson structure is Poisson if one of the following equivalent integrability conditions holds:..."
- Section "Symplectic leaves".
First paragraph: should "completely integrable singular foliation" be a "completely integrable singular distribution"? (a foliation is always integrable)
- Section "Rank of a Poisson structure"
In the sentence "Regular points form an open dense subspace", I think it's better to write "open dense subset".
the sentence "M_reg=M, i.e., \pi^sharp has constant rank" is only correct when M is connected, or it should say "locally constant rank". Perhaps one can just skip the equality "M=M_reg" and write: "When the map \pi^sharp is of constant rank, the Poisson structure \pi is called regular".
- Section "the non-regular case":
Different sign convention relating Poisson bracket and symplectic form (as mentioned above, this one seems the good one to use)
- Section "Trivial Poisson structures", include "\forall f,g \ni C^\infty(M)"
- Section "nondegenerate Poisson structures"
The sentence "Non-degenerate Poisson structures have only one symplectic leaf, namely M itself" again is correct only assuming that M is connected.
I cannot make sense of the sentence "their Poisson algebra .... becomes a Poisson ring"
- Section "Linear Poisson structures",
there is a typo in d_\xi f, d_\xi g : T_\xi \g^* -> R^n (it should be just R).
- Section "Fibrewise linear Poisson structures"
2nd paragraph: denote Lie algebroid by a triple (A,\rho [ , ]).
Paragraph after the 3rd equation in display: "The symplectic leaves of A* are the cotangent bundles of the algebroid orbits O in A" This claim is not correct, it would in particular mean that coadjoint orbits are cotangent bundles... The sentence that follows it: "if A is integrable to a Lie groupoid, they are the connected components..." is fine, but this cannot be "equivalent" to the previous claim.
In the paragraph just below, one could mention, more generally, that any fibrewise linear Poisson structure that is nondegenerate is isomorphic to the canonical symplectic form on T*M.
- Section "Other examples and constructions",
5th bullet point: "foliated two-form" rather than "foliation two-form"
6th bullet point: "Poisson diffeomorphisms" and "Poisson map" are only introduced later... one could say that the bracket on M/G comes from the fact that the Poisson bracket of G-invariant functions on M is G-invariant.
- Section "Modular class".
The definition of modular vector field/class is only given when M is orientable (so one can consider volume forms), so this should be said (the definitions can be formulated in general, one should add refs)
- Section "Poisson maps"
in the 4th bullet point of equivalent definitions, one should be more precise and say "forward Dirac morphism" (since there is also a notion of backward Dirac), and provide a reference for this notion.
in "Examples", second bullet point, "subset" instead of "subspace".
In the last paragraph (before symplectic realisations), one can mention that Poisson maps between symplectic manifolds must be submersions, while symplectic maps must be immersions.
- Section "symplectic realisation".
second paragraph: the reference to "this last condition" is not clear (it has nothing to do with the sentence prior to this reference).
First bullet point below: typo, it should read "and phi as the projection..."
Section "Integration of Poisson manifolds" Include reference to the last sentence "It is of crucial importance to notice...", this was first noticed in ref 25 (further elaborated in 24, 30)
- Section "symplectic groupoid"
2nd paragraph: The claim in the last sentence: "Conversely, if the cotangent bundle T*M of a Poisson manifold is integrable to some Lie groupoid G, then G is automatically a symplectic groupoid.[28]" should be fixed. The only groupoid integrating T*M guaranteed to be symplectic is the *source-simply-connected* integration. (and, by the way, reference 28 seems to have nothing to do with that). This result goes back to mackenzie-Xu in their paper about integration of Lie bialgebroids.
A simple example is the 3-sphere S^3 with the trivial Poisson structure. Its source simply connected integration is T*S^3= S^3 x R^3, which is symplectic. but S^3 \times T^3 is another integration, but admits no symplectic structure (at all).
4th paragraph: last sentence "Using such obstructions, one can show that....", one can actually show that more directly, without using the obstructions (see e.g. D. Alvarez, Proceedings of the AMS 149 (2021))
- section "Examples of integrations".
In every example, it is written "*the* symplectic groupoid being...." but it would be better to say "*a* symplectic groupoid being", unless you want to especifically describe *the* source-simply-connected integration (but then the second and fourth bullet points need more assumptions).
In the third bullet point, to just give one possible integration, there is no need for G to be simply connected (this is in fact a particular case of the 4th bullet point, which has no such assumption).
- Section on "Submanifolds"
The definition of Poisson submanifold in the first paragraph is not well formulated... In principle there is no "\pi|_N", unless you say that N is "tangent to \pi", which is already the definition of Poisson submanifold...you could instead consider N together with a Poisson structure for which the inclusion is a Poisson map. The second definition via hamiltonian vector fields is fine, which is just a way to say that N is tangent to \pi.
OhanaUnitedTalk page 21:10, 11 September 2023 (UTC)
- I thank the reviewer for the thorough and useful comments. I have implemented all of his corrections and suggestions (with one exception - see below); in particular (I will not mention fixing typos in the text or the formulas),
- - I fixed the convention of the Poisson bracket of a symplectic form
- - I rewrote the ambiguous sentences in the 5th and 6th paragraph of the introduction as suggested
- - in "Equivalence of the definition" I rephrased the definition of almost Poisson structure as suggested
- - in "Symplectic leaves" I changed singular foliation into singular distribution
- - in "Rank of a Poisson structure" I added the word open and removed the reference to M_reg = M
- - in "Nondegenerate Poisson structures" I added the connectedness hypothesis and I removed the reference to Poisson rings (I realised it is not a commonly used algebraic notion; actually, I could not find it in any source besides the wikipedia page Poisson ring, which has indeed no reference)
- - in "Fibrewise linear Poisson structures" I fixed the claim on the symplectic leaves of A^* and added the suggested remark
- - in "Other examples and constructions" I replaced "by Poisson diffeomorphisms" with the suggested condition
- - in "Modular class" I added the orientability hypothesis and then explained that it can be removed by replacing volume forms with densities, adding a reference
- - in "Poisson maps" I added the word "forward" to Dirac map, adding a reference; I also added a line involving general Poisson/symplectic maps between symplectic manifolds, as suggested
- - in "Symplectic realisation" I rewrote the sentence which contained "this last condition"
- - in "Integration of Poisson manifolds" I added the appropriate references
- - in "Symplectic groupoid" I added the hypothesis of s-simply connectedness when needed and fixed/updated the references
- - in "Examples of integrations" I replaced "the" with "a" and removed the request of simply connectedness for G
- - in "Submanifolds" I rewrote the definition of Poisson submanifold
- The only exception involves the comment on the opening sentence. While I completely agree with the reviewer that something of the kind "A Poisson manifold is a smooth manifold endowed with a Poisson structure. The notion of Poisson manifold generalises..." would be the most appropriate opening for a paper in a mathematics journal, I believe that the Wikijournal of Science prefers a less mathematical-heavy introduction.
- Indeed, my first sentence is the standard opening for the majority of wikipedia articles involving topics in advance mathematics (compare e.g. with Symplectic manifold or other manifolds with an extra structure): "In X, a field of mathematics, a Y is a Z blablabla", where Z is a more basic notion than Y, which however does not prevent the non-expert reader to finish to read the sentence (and, ideally, the wikilinks would point to the readers what they should be familiars in order to read further).
- I realise however that the difference might be very minor, therefore I would like to ask the editors' opinion: I am open to any reasonable change on this matter.
- PS for the editors: I have also implemented a few minor corrections (basically only typos) to reflect the new edits made in the last months by other users (or citation bots) on the original wikipedia page on which this article is adapted. Francesco Cattafi (discuss • contribs) 17:45, 22 October 2023 (UTC)
Style of the references
I have a further question for the editors (I realised this issue only while implementing some of the reviewer's comments). In the original wikipedia articles I added several wikilinks into each item in the bibliography (basically linking the pages of the authors and of the journals, when available): all such links have been lost when transporting the article from wikipedia to wikiversity.
Similarly, in the articles written in French, the English translation of the title disappeared; last, most of the references had also a link to the Arxiv (pointing to the open access version of those papers), which was removed as well.
I was wondering if this has been intentional, i.e. it is the journal policy to minimize the data of each reference item, or if the missing parts/links have been simply a collateral damage during the transportation of the article, and, therefore, I can/should restore them. Francesco Cattafi (discuss • contribs) 17:58, 22 October 2023 (UTC)
- I don't know. What does the edit history tell you? Physikerwelt (discuss • contribs) 07:49, 18 June 2024 (UTC)
- @Francesco Cattafi and @Physikerwelt, I have narrowed it down to a a series of edits in March 2023 which removed "|author-link=" to the respective references. These edits took out the wikilinks to the authors and journal names. In my opinion, it's non-standard to link to the Wikipedia page of specific authors or journals. Not sure about the French articles. Can you point to which reference that is in French? As far as I can tell, Arxiv is still embedded in {{cite journal}}. However, I think the template supersedes the Arxiv field if URL field is filled. OhanaUnitedTalk page 19:31, 18 June 2024 (UTC)
- @OhanaUnited thank you for investing. This explains what happened.
- @Evolution and evolvability what was the rational to remove links and author references? Physikerwelt (discuss • contribs) 06:40, 19 June 2024 (UTC)
- @Francesco Cattafi and @Physikerwelt, I have narrowed it down to a a series of edits in March 2023 which removed "|author-link=" to the respective references. These edits took out the wikilinks to the authors and journal names. In my opinion, it's non-standard to link to the Wikipedia page of specific authors or journals. Not sure about the French articles. Can you point to which reference that is in French? As far as I can tell, Arxiv is still embedded in {{cite journal}}. However, I think the template supersedes the Arxiv field if URL field is filled. OhanaUnitedTalk page 19:31, 18 June 2024 (UTC)
- Thank you all for the replies. Regarding the author and journal links, I was following the Cite Journal template, which has indeed an author-link parameter (so I assumed it should be used, when possible) and states, regarding the journal, "may be wikilinked".
- You are right regarding the Arxiv links, it seems that it is simply not displayed in wikiversity, as opposed to wikipedia. The articles in French are those referenced as 1, 4, 13, 16 and 28. Francesco Cattafi (discuss • contribs) 19:37, 26 June 2024 (UTC)
Second Review
Review by Eckhard Meinrenken , University of Toronto
These assessment comments were submitted on , and refer to this previous version of the article
The article is generally well written, and there are no mistakes. I just have a few comments regarding references and choice of topics.
1. "Symplectic realization" is in my opinion a rather specialized topic. I see this as a technical tool towards symplectic groupoids; I wouldn't devote so much space to them.
2. "Weinstein groupoid" is perhaps a bit of a misnomer. A similar construction was suggested by Severa in his talk at Poisson 2000, also sketched in his "letters to Weinstein". See e.g. Severa's article https://math.uni.lu/travaux/Last/7SEVERA.PDF Maybe call it a Severa-Weinstein groupoid.
3. In the section on submanifolds, the claim that "Poisson submanifolds are rare" since there are none for a symplectic manifold (and using this to motivate Poisson transversals) seems unconvincing. Consider the opposite extreme of a zero Poisson structure; here every submanifold is a Poisson submanifold but the only Poisson transversals are the open subsets. It might be good to give some examples of Poisson submanifolds: symplectic leaves; or the spheres in side the dual of a Lie algebra of a compact Lie group. I would also say that coisotropic submanifolds are extremely important (the graph of a Poisson map for example; cf. Weinstein's coisotropic calculus).
4. It might be good to give some simple examples of Poisson structures; e.g. in coordinates.
5. Some topics are omitted:
a) Deformation quantization? (One might says it's one of the major themes and/or motivations of Poisson geometry.)
b) Linearizability questions? Conn's theorem?
c) Poisson Lie groups.
- I thank the reviewer for the thorough and useful comments; I have implemented all of his suggestions.
- 1) I have reduced the section on symplectic realisations and moved it the end of the section on integration of Poisson manifolds. In particular, I have removed the technical discussions on the adjective "full" and the three examples (which were indeed redundant, since they are the same examples already discussed in the integration of Poisson manifolds). Now only a couple of paragraphs are left, simply explaining the relations between (complete) symplectic realisations and integrability.
- 2) I renamed the Weinstein groupoid as Severa-Weinstein groupoid and added the appropriate reference. Notice however that this article has a link to the wikipedia page Lie algebroid, where such object is discussed still under the name Weinstein groupoid - I will soon update that page as well with the more complete name and reference. Edit: I have added the name Severa-Weinstein groupoid also in that page, updating the wikilink.
- 3) I removed the unconvincing motivation which linked Poisson submanifolds to Poisson transversal and improved the exposition on submanifolds. Now there is a section on Poisson submanifolds, followed by a list of examples, and then a section on "other submanifolds in Poisson geometry", which contains the previous paragraph on Poisson transversals, as well as a new paragraph on coisotropic submanifolds.
- 4) I have added in the section "Linear Poisson structures" the explicit expression in coordinates for two simple Lie-Poisson structures, namely those of so(3,R)^* and sl(2,R)^*. Moreover, I added another simple example in coordinates of a log-symplectic structures (another important class of Poisson manifolds which was not mentioned elsewhere in the article).
- 5) I have added a section "Further topics" covering deformation quantisation, the linearisation problem and Poisson-Lie groups. Notice that I have added a "see also" template linking to the existing wikipedia page on Poisson-Lie groups, where they are treated in more details (unlike the other two topics, which currently do not have a separate page - even if they probably would deserve one).