Definition

For a d-dimensional Poisson manifold , at a point , the rank is simply the rank of the structure matrix at .

With a bit more detail, given a point and coordinates on a neighborhood of the Poisson bivector can be expressed in the coordinates as

Taking the matrix we have the structure matrix.

The rank at is the rank of the matrix .

The rank of is the maximum

The rank is said to be maximal if .

Relation to Hamiltonian directions

For a Poisson manifold , let .

is even and is equal to .
For every , the subset of , defined by

is open.

Proof

todo - Get from Poisson structures book…

References

@laurent-gengoux2012 - Chapter 1.3