nLab Poisson tensor

Definition

For $(X, \{-,-\})$ a Poisson manifold, there is a unique bivector (skew-symmetric rank (2,0)-tensor field) $\pi \in \wedge^2 \Gamma(T X)$ such that for all functions $f,g \in C^\infty(X)$ the Poisson bracket is given by the Schouten bracket $[-,-]$ as

$$\{f,g\} = [[\pi, f],g] \,.$$

This $\pi$ is called the Poisson tensor or Poisson bivector of $(X, \{-,-\})$.

Every bivector $\pi$ such that $[\pi, \pi] = 0$ in the Schouten bracket arises this way.

In terms of Lie algebroids

The Poisson tensor constitutes the anchor map of the Poisson Lie algebroid $\mathfrak{P}$ which corresponds to the Poisson manifold.

Regarded as an element in the Chevalley-Eilenberg algebra $CE(\mathfrak{P}) \simeq (\wedge^\bullet \Gamma(T X), [\pi,-])$, the Poisson tensor also constitutes the canonical Lie algebroid cocycle on $\mathfrak{p}$ which is in transgression with the canonical invariant polynomial on $\mathfrak{P}$, the one that exhibits $\mathfrak{P}$ as a symplectic Lie 1-algebroid.

Related concepts

• deformation quantization

  • Moyal quantization