My primary motivation for asking this question comes from the discussion taking place in the comments to [What is a symplectic form intuitively?](http://mathoverflow.net/questions/19932/).

Let $M$ be a smooth finite-dimensional manifold, and $A = \cal C^\infty(M)$ its algebra of smooth functions.  A _derivation_ on $A$ is a linear map $\{\}: A \to A$ such that $\{fg\} = f\{g\} + \{f\}g$ (multiplication in $A$).  Recall that all derivations factor through the de Rham differential, and so: __Theorem:__ Derivations are the same as vector fields.

A _biderivation_ is a linear map $\{,\}: A\otimes A \to A$ such that $\{f,-\}$ and $\{-,f\}$ are derivations for each $f\in A$.  By the same argument as above, biderivations are the same as sections of the tensor square bundle ${\rm T}^{\otimes 2}M$.  _Antisymmetric_ biderivations are the same as sections of the exterior square bundle ${\rm T}^{\wedge 2}M$.  A _Poisson structure_ is an antisymmetric biderivation such that $\{,\}$ satisfies the Jacobi identity.

Recall that sections of ${\rm T}^{\otimes 2}M$ are the same as vector-bundle maps ${\rm T}^*M \to {\rm T}M$.  A _symplectic structure_ on $M$ is a Poisson structure such that the corresponding bundle map is an isomorphism.  Then its inverse map makes sense as an antisymmetric section of ${\rm T^*}^{\otimes 2}M$, i.e. a differential 2-form, and the Jacobi identity translates into this 2-form being closed.  So this definition agrees with the one you may be used to of "closed nondegenerate 2-form".

> __Question:__ Is there a "purely algebraic" way to test whether a Poisson structure is symplectic?  I.e. a way that refers only to the algebra $A$ and not the manifold $M$?

For example, it is necessary but not sufficient that $\{f,-\} = 0$ implies that $f$ be locally constant, where I guess "locally constant" means "in the kernel of every derivation".  The easiest way that I know to see that it is necessary is to use Darboux theorem to make $f$ locally a coordinate wherever its derivative doesn't vanish; it is not sufficient because, for example, the rank of the Poisson structure can drop at points.

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