The Poisson Bivector and the Schouten-Nijenhuis Bracket

Abstract

In this chapter we shall discuss various ways of representing a Poisson structure on a differentiable manifold. Let Mⁿ be a Poisson manifold. The basic remark is the following obvious consequence of (0.4): {f,.} is a derivation of C^∞(M). Hence ∀f ∈ C∞(M) there exists a well defined vector field X_f such that

$$\{f,g\}= X_f g = - X_g f = dg(X_f) = -df(X_g)$$

(1.1)

. X_f will be called the Hamiltonian vector field of f.