The Full Wiki

Poisson algebra: Wikis

Advertisements

Note: Many of our articles have direct quotes from sources you can cite, within the Wikipedia article! This article doesn't yet, but we're working on it! See more info or our list of citable articles.

Encyclopedia

From Wikipedia, the free 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.

Contents

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.

Advertisements

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.

See also

References


Advertisements






Got something to say? Make a comment.
Your name
Your email address
Message