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.

ymplectic 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 $\{a,b\}=a\_0b$ and $a\; cdot\; b\; =a\_\{-1\}b$. For certain vertex operator algebras, these Poisson algebras are finite dimensional.

ee also

*Poisson superalgebra
*Antibracket algebra

References

*springer|id=p/p110170|title=Poisson algebra|author=Y. Kosmann-Schwarzbach