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This question is (at least as I read it) about the Poisson bracket; the Poisson bracket is a Lie bracket structure on the functions on a symplectic manifold. So how should one think about Poisson b …

Obviously, this is a question that could be interpreted in different ways. For me, Darboux's theorem is the symplectic analogue of the theorem that a flat Riemannian manifold (i.e. one where Riemann' …

Since the earlier (very nice) answers didn't actually say it, let me mention: lots of exact symplectic manifolds are not cotangent bundles. The hypersurface $xy=f(z)$ for $f$ a monic polynomial with …

A more general form of Francesco's answer is that coisotropic submanifolds are those locally defined as the zero set of some Poisson commuting functions, and Lagrangians are those where the number of …

Yes. $T^2 \times T^2$ with the sum of the volume forms on each factor.

One thing you didn't mention is the manifold in which this calculation happens: the Slodowy slice to a nilpotent of type $(n,n)$. This manifold is the key to everything, since it is a geometric avatar …

One thing that seems to be confusing you is that there is no "the moment map." There often several different moment maps for the same action, and as Jose says, sometimes none. On the other hand, you …

Nowhere. The paper is still in preparation, and looks to be for a few more months at least. Probably the best document at the moment is this (extremely long) set of talk slides of mine. I should no …

The moral answer is "Yes. A Poisson structure is symplectic if and only if the algebra has no interesting Poisson ideals." The idea is this: a Poisson ideal is one which is closed under the operatio …

In this case, if you pick a $S^1$-invariant metric, the Morse flow will commute with the $S^1$ action, and so you can think of them together as an action of $\mathbb{C}^*$ where the unit circle acts b …

Work of Gan and Ginzburg (https://arxiv.org/pdf/math/0409262v7.pdf) shows that one of our favorite examples, $M=T^*(\mathfrak{gl}_n\oplus \mathbb C^n)$ and $G=GL(n)$, has a 0 level of the moment map w …

Well, there are stupid examples like the fact that $\mathbb{P}^n$ has Kähler structures where any rational multiple of the hyperplane class is the Kähler class which are compatible with the standard …

I believe you should interpret $dX\wedge dY$ as follows: $X$ is a matrix valued function on the space of pairs of matrices (which just takes the first one); its entries are honest functions $x_{i,j}$. …

We can assume that $G$ is affine, since an abelian variety must act trivially on any affine variety. The closed points of $X$ are exactly the closed $G$-orbits on $\mu^{-1}(0)$. On an affine variety, …

In my paper "Singular blocks of parabolic category O and finite W-algebras", these are called "S3-varieties." S3 is for "Slodowy-Springer-Spaltenstein."