nLab
symplectic realization

Contents

Contents

Definition

For (X,π)(X, \pi) a Poisson manifold, a symplectic realization of it is a symplectic manifold (Y,ω)(Y, \omega) and a Poisson map (Y,ω)(X,π)(Y, \omega) \to (X, \pi) such that YXY \to X is a surjective submersion YXY \to X.

Properties

Solution in terms of symplectic groupoids

For any symplectic groupoid Σ\Sigma with base a Poisson manifold PP the target map is a symplectic realization of PP and the source map is a symplectic realization of the opposite structure. Thus Σ\Sigma with its symplectic structure may be regarded as a desingularization of PP with its Poisson structure. Since the symplectic groupoid is the Lie integration of the Poisson Lie algebroid of the Poisson manifold, symplectic realization has been reduced to a problem in Lie theory.

Created on February 12, 2013 at 21:14:46. See the history of this page for a list of all contributions to it.