Poisson algebra

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 and Lie algebras

If A is an associative algebra, then imposing the commutator [x,y]=xy−yx turns it into a Poisson algebra. A very explicit construction of this is given on the article on universal enveloping algebras: In that particular case, one starts with a Lie algebra, for which the commutator is already defined. However, one could equally well start with any unital associative algebra, and impose the commutator to obtain the corresponding Lie algebra A_L (as a quotient space, of the original algebra modulo the commutator).

The construction proceeds by first building the tensor algebra of the Lie algebra. One can then show that the commutator can be consistently lifted to the entire tensor algebra: it obeys both the product rule, and the Jacobi identity of the Poisson bracket, and thus is the Poisson bracket, when lifted. As a result, one has the general statement that the tensor algebra of any Lie algebra is a Poisson algebra. The universal enveloping algebra is obtained by modding out the Poisson algebra structure.

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.

See also

• Poisson superalgebra
• Antibracket algebra
• Moyal bracket
• Kontsevich quantization formula

References

• Y. Kosmann-Schwarzbach (2001) [1994], "Poisson algebra", Encyclopedia of Mathematics, EMS Press
• Bhaskara, K. H.; Viswanath, K. (1988). Poisson algebras and Poisson manifolds. Longman. ISBN 0-582-01989-3.