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:

\{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 noncommutative associative algebra, then the commutator [x,y]≡xy−yx turns it into a Poisson algebra.

Vertex operator algebras

For a vertex operator algebra $(V,Y,\omega ,1)$ , the space $V/C_{2}(V)$ is a Poisson algebra with $\{a,b\}=a_{0}b$ and $a\cdot b=a_{-1}b$ . For certain vertex operator algebras, these Poisson algebras are finite dimensional.