Poisson algebra

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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.

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[edit] 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, 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.

[edit] Examples

Poisson algebras occur in various settings.

[edit] 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.

[edit] Associative algebras

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

[edit] 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.

[edit] See also

[edit] References

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