Poisson manifold

As a Poisson manifold is called in mathematics a differentiable manifold M, which is provided with a bilinear algebra of smooth functions on M which satisfies the properties of a Poisson bracket (named after Siméon Denis Poisson ).

Definition

A Poissonstruktur on M is a bilinear map

So that the bracket is antisymmetric:

The Jacobi identity is enough:

And a derivation is:

For all

Examples and Applications

In particular, any symplectic manifold is a Poisson manifold. In this case, the defining structure

Given by a 2- form or their components in the local coordinate.

Poisson manifolds can be viewed as algebraic abstractions of symplectic manifold. Differences exist with a much larger class of morphisms then for example that the condition is dropped, the Poisson bracket should never be singular, so have full rank.

Applies this calculation, for example in the deformation theory. It provides access to non-commutative geometry there and geometric quantization.

• Differential Geometry
• Plurality