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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Definition
A Poisson algebra is a vector space over a field K equipped with two bilinear products, <math>\cdot</math> and [ , ], having the following properties:
- The product <math>\cdot</math> 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, 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 <math>H</math> on the manifold induces a vector field <math>X_H</math>, the Hamiltonian vector field. Then, given any two smooth functions <math>F</math> and <math>G</math> over the symplectic manifold, the Poisson bracket {,} may be defined as:
- <math>\{F,G\}=dG(X_F)\,</math>.
This definition is consistent in part because the Poisson bracket acts as a derivation. Equivalently, one may define the bracket {,} as
- <math>X_{\{F,G\}}=[X_F,X_G]\,</math>
where [,] is the Lie derivative. When the symplectic manifold is <math>\mathbb R^{2n}</math> with the standard symplectic structure, then the Poisson bracket takes on the well-known form
- <math>\{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}.</math>
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 alebras
For a vertex operator algebra <math>(V,Y, \omega, 1)</math>, the space <math>V/C_2(V)</math> is a Poisson algebra with <math>\{a,b\}=a_0b</math> and <math>a \cdot b =a_{-1}b</math>. For certain vertex operator algebras, these Poisson algebras are finite dimensional.