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Concept
Poisson algebra
Summary
In mathematics, a Poisson algebra is an associative algebra together with a Lie bracket that also satisfies Leibniz's 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 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.
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 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
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
Similar considerations apply for Poisson manifolds, which generalize symplectic manifolds by allowing the symplectic bivector to be rank deficient.
The tensor algebra of a Lie algebra has a Poisson algebra structure. A very explicit construction of this is given in the article on universal enveloping algebras.
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