nLab
Poisson tensor

Contents

Contents

Definition

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

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

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

Every bivector π\pi such that [π,π]=0[\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 𝔓\mathfrak{P} which corresponds to the Poisson manifold.

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

Last revised on April 2, 2013 at 20:40:55. See the history of this page for a list of all contributions to it.