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.

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, y ⋅ z} = {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 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:

$\{F,G\}=dG(X_{F})=X_{F}(G)\,$ .

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 R²ⁿ 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 an 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, ω, 1), the space V/C₂(V) is a Poisson algebra with {a, b} = a₀b and a ⋅ b = a₋₁b. For certain vertex operator algebras, these Poisson algebras are finite dimensional.

References

Y. Kosmann-Schwarzbach (2001), "Poisson algebra", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Bhaskara, K. H.; Viswanath, K. (1988). Poisson algebras and Poisson manifolds. Longman. ISBN 0-582-01989-3.

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