Poisson manifold

Overview

A Poisson manifold is a generalization of symplectic manifolds. It is a way to describe families of phase spaces of physical systems.

There are two ways to define a Poisson manifold. One uses the Poisson bracket on the algebra of smooth functions . The other uses bivector fields.

Poisson bracket definition

A Poisson manifold is a smooth manifold and a Poisson bracket on the algebra of smooth functions

A morphism in the category of Poisson manifolds is a Poisson map.

Bivector field definition

A Poisson manifold is a smooth manifold with a bivector fields satisfying where denotes the Schouten bracket.

Intuition behind conditions

For a bivector field , we can look at

Since , then we know that the bracket is an anti-symmetric derivation. That takes care of the antisymmetric and Liebneiz rule condition on the Poisson bracket given the definition above.

Why do we need the condition ?

Let , using definition of the Schouten bracket we can look at for functions

In order to compute we can see that there are only 3 (1,2)-shuffles.

Therefore, letting , using the second part of the explicit construction of the Schouten bracket we have

with the Poisson bracket defined above.

Therefore,

So the only way a derivation induced from a bivector field will satisfy the Jacobi identity is if and only if .