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I have added the relations
coisotropic submanifold $\leftrightarrow$ Lagrangian submanifold in Poisson Lie algebroid
Dirac structure $\leftrightarrow$ Lagrangian submanifold in Courant Lie 2-algebroid
to Lagrangian submanifold and cross-linked with various related entries, such as polarization.
sorry: Lagrangian submanifold
added to Lagrangian submanifold in the section Of a Poisson Lie algebroid the argument showing that Lagrangian dg-submanifolds/sub-Lie algebroids of Poisson Lie algebroids are precisely co-isotropic submanifolds of the underlying Poisson manifolds.
Neat. But the table is not logical to me. The table says symplectic manifold:Lagrangean submanifold is like Poisson Lie algebroid:coisotropic submanifold, what is not quite what is said in 4 nor in the entry (“A Poisson manifold induces a Poisson Lie algebroid, which is a symplectic Lie n-algebroid for n=1. Its coisotropic submanifolds correspond to the Lagrangian dg-submanifolds (see there) of this Poisson Lie algebroid.”) Coisotropic manifold is of an object (Poisson manifold) which is not mentioned in the second line, while Lagrangean submanifold is of the symplectic manifold, what is not the same relation then.
I have now expand the relevant item in the table from just
“coisotropic submanifold”
to
“coisotropic submanifold (of the underlying Poisson manifold)”
Is that better?
The thing is that both Poisson manifolds and Poisson Lie algebroids as well as coisotopic submanifolds of the former and Lagrangian sub-Lie algebroids of the latter are not exactly the same, but are in canonical correspondence to each other, which makes us speak about one as about the other. It’s an abuse of terminology, but a convenient one that seems useful for the brevity of a table.
Well, this was not a problem, the abuse of language was a problem; one would expect that the thing is defined in the same way if straightforwardly reading the table. So, I would say what it is – the dg Lagrangian sub-Lie algebroid and then have a separate column to what it corresponds to after some duality. Otherwise, without a new column, one should say the dual of … coisotropic submanifold. This would not confuse, as it would be completely parallel construction. But never mind if it is not convenient.
Hm, there is no dualization going on. Probably I am misunderstanding what you mean.
The statement expressed by the table is supposed to be this:
A Lagrangian sub-Lie algebroid = Lagrangian dg-submanifold of a Poisson Lie algebroid is equivalently a coisotropic submanifold of the corresponding Poisson manifold.
Oh, thanks, I am getting it now, the correspondence is rather trivial, not a duality. It is good the way it is :)
I created a new stub Poisson cohomology on a topic which is, I guess, in the Lie algebroid phrasing (more references, MacKenzie?) related to the issues in this thread.
Yes, just last week Johannes Huebschmann showed that Poisson cohomology is nothing but the Lie algebroid cohomology of Poisson Lie algebroids:
I have added that reference to Poisson cohomology and added an Idea-paragraph from this natural perspective.
Well, Weinstein and da Silva say essentially this in 18.4 of their 1999 monograph. Hence it is not a new fact.
Oh, here we are. Huebschmann’s article is from 1990, at arXiv one has just a reprint of his Crelle 1990 article ! Weinstein 1999 follows Huebschmann 1990! I will add the reference.
Edit: the abstract of the arXiv preprint version is added and has not appeared in the original 1990 paper.
Oh, thanks! I had missed that.
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