For a Poisson manifold, there is a unique bivector (skew-symmetric rank (2,0)-tensor field) such that for all functions the Poisson bracket is given by the Schouten bracket as
This is called the Poisson tensor or Poisson bivector of .
Every bivector such that in the Schouten bracket arises this way.
The Poisson tensor constitutes the anchor map of the Poisson Lie algebroid which corresponds to the Poisson manifold.
Regarded as an element in the Chevalley-Eilenberg algebra , the Poisson tensor also constitutes the canonical Lie algebroid cocycle on which is in transgression with the canonical invariant polynomial on , the one that exhibits 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.