Definitions

Poisson algebra

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 product $cdot$ forms an associative K-algebra.
• The product { , }, called the Poisson bracket, forms a Lie algebra, and so it is anti-symmetric, and obeys the Jacobi identity.
• 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, yz} = {x, y}z + y{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 $X_H$, the Hamiltonian vector field. Then, given any two smooth functions $F$ and $G$ over the symplectic manifold, the Poisson bracket {,} may be defined as:

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

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

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

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

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

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

Vertex operator algebras

For a vertex operator algebra $\left(V,Y, omega, 1\right)$, the space $V/C_2\left(V\right)$ is a Poisson algebra with $\left\{a,b\right\}=a_0b$ and $a cdot b =a_\left\{-1\right\}b$. For certain vertex operator algebras, these Poisson algebras are finite dimensional.