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, 
and { , }, having the following properties:
- The product 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 , 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:
.
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]\,](../../../../math/b/a/0/ba0d9927fc77e849056be11d37903869.png)
where [,] is the Lie derivative. When the symplectic manifold is 
with the standard symplectic structure, then the Poisson bracket takes on the well-known form
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 
. For certain vertex operator algebras, these Poisson algebras are finite dimensional.
[edit] See also
[edit] References
- Y. Kosmann-Schwarzbach (2001), “Poisson algebra”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
