# nLab symplectic realization

Contents

### Context

#### Symplectic geometry

symplectic geometry

higher symplectic geometry

# Contents

## Definition

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

## Properties

### Solution in terms of symplectic groupoids

For any symplectic groupoid $\Sigma$ with base a Poisson manifold $P$ the target map is a symplectic realization of $P$ and the source map is a symplectic realization of the opposite structure. Thus $\Sigma$ with its symplectic structure may be regarded as a desingularization of $P$ 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.