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.

A Poisson algebra is a vector space over a field K equipped with two bilinear products, ⋅ and {, }, having the following properties:

The last property often allows a variety of different formulations of the algebra to be given, as noted in the examples below.

Poisson algebras occur in various settings.

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:

This definition is consistent in part because the Poisson bracket acts as a derivation. Equivalently, one may define the bracket {,} as

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

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