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.


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.

Symplectic manifolds[edit]

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 vanishing on some (or trivially, all) of the manifold.

Associative algebras[edit]

If A is an associative algebra, then the commutator [x,y]≡xyyx turns it into a Poisson algebra.

Vertex operator algebras[edit]

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

See also[edit]