Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
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.
Added:
The original reference is
added pointer to:
Is there a smooth moduli stack of Poisson structures?
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.
Hmm, I see, thanks, Urs.
This would be a great example to spell out a little more on the nLab!
1 to 7 of 7