Yesterday I discussed the Poisson bracket on a symplectic manifold and some of its basic properties. Naturally enough, someone decided to axiomatize all this.

**Poisson manifolds **

So, a **Poisson manifold** is a smooth manifold together with a Lie algebra structure (the Poisson bracket) on the space of smooth functions on such that

To check that a symplectic manifold is indeed a smooth manifold, we need only recall that there is a vector field with , so the above is just the derivation identity. All the other properties of the Poisson bracket on a symplectic manifold were established yesterday.

Now let’s switch to the general Poisson manifold case.

It turns out that the Poisson structure alone is enough to show many similarities. Indeed, the derivation identity above implies that there is a vector field, still denoted by , with

and in particular it follows that by antisymmetry. The set of such vector fields will be called, as in the symplectic case, **Hamiltonian vector fields**. I claim that the identity

is still true. In the previous post we proved this directly in the symplectic case, and used it to deduce the Jacobi identity, but this time we will go in the opposite direction. The Jacobi identity implies for :

So we get a Lie isomorphism from to the Lie algebra of Hamiltonian vector fields with the usual Lie bracket.

**The cosymplectic structure **

I now claim that it is possible to take a Poisson structure, and turn it into a **cosymplectic structure**. That is, we have a tensor such that is an antisymmetric bilinear map on each cotangent space . Note that is not required to be nondegenerate. This terminology admittedly seems confusing.

Note that a symplectic structure (which is required to be nondegenerate!) induces a cosymplectic structure by the natural isomorphisms from duality. Anyway, to construct this structure, we take

which at first appears ambiguous, but is in fact seen to be well-defined. What we have to do is prove that if are smooth functions with the same differential at the point , then

The analogous identity in the other variable follows by antisymmetry. But this follows because

**Morphisms of Poisson manifolds **

A **Poisson map** between Poisson manifolds is one that commutes with the Poisson structures. In other words, if are smooth on , then

i.e. that the map is a Lie algebra homomorphism. Perhaps there is an analogy here to the notion of -relatedness. Vector fields on respectively are said to be -related if for every ,

It is a basic fact that if are respectively -related to , then is -related to . Here it is functions that are related by and the Poisson bracket that is preserved. (Ok, maybe this was a bit much, but I am a fan of analogies.)

**Products of Poisson manifolds **

Given Poisson manifolds , I claim that there is a unique way to define a Poisson structure on the product manifold such that the projections are Poisson maps. Moreover, we have injections of Lie algebras

We want the images to commute. So, we first define the Poisson bracket on —it is the product Lie algebra. Then we note that given and , we can choose such that and have the same differential at . Moreover the differentials of at are uniquely determined by this. Thus, we use this near-splitting of into to get the full Poisson structure on the product.

**The splitting theorem **

First, we define the rank of the Poisson structure at a point . This is the dimension of the largest subspace of such that the bilinear form is nondegenerate.

The splitting theorem, due to Weinstein, describes Poisson manifolds locally:

Theorem 1 (Splitting theorem)Let be a Poisson manifold and . Then there is an open neighborhood of Poisson-isomorphic to a product for symplectic and Poisson of rank zero at (the inverse image of) .

This will be proved by induction on . When , the form is identically zero, and the result is immediate.

Assume the theorem proved for dimensions smaller than . Now there are two cases. One is that the form vanishes at , in which case we can take .

The more interesting one is that does not vanish on . We consider this next. So we can find a smooth function with

We can straighten out by choosing an appropriate coordinate system so that . Then writing , we have

We are going to use to obtain a new, better system of local coordinates. First, and are linearly independent in some neighborhood of because . Also

so we can choose a set of local coordinates with

I now claim that are also a system of local coordinates. Indeed,

This system of coordinates has the important property that

because these equal respectively . Let the corresponding coordinate neighborhood be , where corresponds to the being fixed and to . Then are open neighborhoods in euclidean spaces.

Now have Poisson structures on them; indeed, they are determined by the values on the coordinate system. The “important property” above shows that commute.

Moreover, on we have the Poisson structure associated to the usual symplectic structure on . This is because the brackets on the local coordinates determine the whole structure, again.

Now we apply the inductive hypothesis to , and take the product of with the symplectic portion of (or some neighborhood thereof) as the , and the other factor as .

There is a uniqueness theorem as well, which I may or may not get to tomorrow.

September 8, 2012 at 8:42 pm

There is a typo in the section ‘Morphisms of Poisson Manifolds’. In the last paragraph, the first $ Y_1 $ that appears should be changed to $ X_2 $.

September 16, 2012 at 9:41 am

Fixed, thanks.