# Poisson algebra: Wikis

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# Encyclopedia

In mathematics, a Poisson algebra is an associative algebra together with a Lie bracket that also satisfies Leibniz' law; that is, the bracket is also a derivation. Poisson algebras appear naturally in Hamiltonian mechanics, and are also central in the study of quantum groups. Manifolds with a Poisson algebra structure are known as Poisson manifolds, of which the symplectic manifolds and the Poisson-Lie groups are a special case. The algebra is named in honour of Siméon-Denis Poisson.

## Definition

A Poisson algebra is a vector space over a field K equipped with two bilinear products, $\cdot$ and { , }, having the following properties:

• The Poisson bracket acts as a derivation of the associative product $\cdot$, so that for any three elements x, y and z in the algebra, one has {x, y$\cdot$ z} = {x, y}$\cdot$ z + y$\cdot${ x, z}.

The last property often allows a variety of different formulations of the algebra to be given, as noted in the examples below.

## Examples

Poisson algebras occur in various settings.

### Symplectic manifolds

The space of real-valued smooth functions over a symplectic manifold forms a Poisson algebra. On a symplectic manifold, every real-valued function H on the manifold induces a vector field XH, the Hamiltonian vector field. Then, given any two smooth functions F and G over the symplectic manifold, the Poisson bracket {,} may be defined as:

$\{F,G\}=dG(X_F)\,$.

This definition is consistent in part because the Poisson bracket acts as a derivation. Equivalently, one may define the bracket {,} as

$X_{\{F,G\}}=[X_F,X_G]\,$

where [,] is the Lie derivative. When the symplectic manifold is $\mathbb R^{2n}$ with the standard symplectic structure, then the Poisson bracket takes on the well-known form

$\{F,G\}=\sum_{i=1}^n \frac{\partial F}{\partial q_i}\frac{\partial G}{\partial p_i}-\frac{\partial F}{\partial p_i}\frac{\partial G}{\partial q_i}.$

Similar considerations apply for Poisson manifolds, which generalize symplectic manifolds by allowing the symplectic bivector to be vanishing on some (or trivially, all) of the manifold.

### Associative algebras

If A is a associative algebra, then the commutator [x,y]≡xyyx turns it into a Poisson algebra.

### Vertex operator algebras

For a vertex operator algebra (V,Y,ω,1), the space V / C2(V) is a Poisson algebra with {a,b} = a0b and $a \cdot b =a_{-1}b$. For certain vertex operator algebras, these Poisson algebras are finite dimensional.