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Results tagged with poisson-geometry
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user 12482
Poisson geometry is the study of varieties endowed with a Poisson structure, which is a certain kind of 2-tensor. This is closely related to symplectic geometry.
1
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Convergence for a family of poisson structures
In general, this seems not to be possible. Consider the case of a vanishing Poisson structure $\Pi = 0$. Then every bivector field $\Lambda$ is closed for the differential $\delta$, which is just the …
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5
votes
Accepted
Which commutative algebras admit a nonzero Poisson bracket?
In general, I think the question is too broard to expect some reasonable answer. But there are many examples and constructions of Poisson brackets which might be interesting for you.
1.) Whenever you …
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8
votes
Accepted
Quantization and noncommutative deformations
Well, a lot of questions, some of which Theo already answered in a very nice way. Let me just give some additional remarks and hints how I think about DQ and Poisson geometry in relation to quantum ph …
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8
votes
Accepted
Some elementary questions about deformation quantization
a lot of questions, let me try on some of them :)
The bad news is that in most of the interesting situations the higher order terms of the star product, the $B_i$ will not vanish. Heuristically this …
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2
votes
Accepted
Poisson structure on the dual Lie algebroid
Depending on you sign convention, this goes as follows. First you denote the bundle projection by $pr\colon E^* \longrightarrow X$. For a section $s \in \Gamma^\infty(E)$ you have a linear function $J …
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1
vote
How to compute the deformation quantizations of a polynomial Poisson algebra?
Not a complete answer but some observations. First it might be necessary not to take formal power series but formal Laurent series in order to get a reasonable behaviour of the Hochschild cohomology. …
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41
votes
What is a Lagrangian submanifold intuitively?
Everything is a lagrangian submanifold (A. Weinstein's lagrangian creed...)
Indeed, every manifold is the lagrangian zero section of its cotangent bundle ;)
But more serious: Weinstein's tubular neig …
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4
votes
Interpretation of the Schouten bracket as an integrability condition
One well-known example is the case of bivector fiels $\pi$. Then $[\pi, \pi] = 0$ is equivalent to say that $\{f, g\} = \pi(df, dg)$ is a Poisson bracket, i.e. satisfies the Jacobi identity. In this …
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5
votes
3
answers
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Differential Hochschild Cohomology, general tools?
Background: in deformation quantization one wants to construct formal associative deformations (the star products $\star$) of the algebra of smooth complex-valued functions on a Poisson manifold $M$ i …
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8
votes
Accepted
In the dictionary between Poisson and Quantum, what corresponds to Coisotropic?
Concerning your first question I have a couple of suggestions: first coisotropic is in some sense the best we can have in a truely Poisson situation: there is nothing like lagrangian (unfortunately).
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