Poisson Manifold - The Poisson Bivector

The Poisson Bivector

Given a Poisson manifold (M,{⋅,⋅}_M), the pairing between the cotangent and tangent bundles yields a bivector field η on M, called the Poisson bivector field. The Poisson bivector field is a contravariant skew-symmetric 2-tensor field that satisfies the following:

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Conversely, given a smooth bivector field η on M, we can use the formula above to define a skew-symmetric bracket {⋅,⋅}_η that obeys Leibniz's rule with respect to each argument. However, we cannot claim that {⋅,⋅}_η is a Poisson bracket because the Jacobi Identity may not hold (in this case, we call {⋅,⋅}_η an almost-Poisson structure). Indeed, {⋅,⋅}_η is a Poisson bracket if and only if the Schouten–Nijenhuis bracket is zero.

In terms of local coordinates, the bivector field at a point x = (x₁,...,x_m) can be expressed as

,

so that

.

For a symplectic manifold (M,ω), we can define a bivector field η on M using the pairing between the cotangent and tangent bundles given by the symplectic form ω. This pairing is well-defined because ω is nondegenerate. Hence, the difference between a symplectic manifold and a Poisson manifold is that the symplectic form is regular (of full rank) everywhere but the Poisson bivector field need not have full rank everywhere. When the Poisson bivector field is zero everywhere, we call it the trivial Poisson structure.