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 PoissonLie groups are a special case. The algebra is named in honour of SiméonDenis Poisson.
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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 realvalued smooth functions over a symplectic manifold forms a Poisson algebra. On a symplectic manifold, every realvalued 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 with the standard symplectic structure, then the Poisson bracket takes on the wellknown 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.
If A is a associative algebra, then the commutator [x,y]≡xy−yx turns it into a Poisson algebra.
For a vertex operator algebra (V,Y,ω,1), the space V / C_{2}(V) is a Poisson algebra with {a,b} = a_{0}b and . For certain vertex operator algebras, these Poisson algebras are finite dimensional.
