Poisson algebra

A Poisson algebra is an associative algebra together with a Lie bracket, satisfying Leibniz' law. More precisely, a Poisson algebra is a vector space over a field K equipped with two bilinear products, \cdot and [,] such that \cdot forms an associative K-algebra and [,], called the Poisson bracket, forms a Lie algebra, and for any three elements x, y and z, [x, yz] = [x, y]z + y[x, z] (i.e. the Poisson bracket acts as a derivation).

Examples

  1. The space of smooth functions over a symplectic manifold.
  2. If A is a noncommutative associative algebra, then the commutator [x,y]≡xy−yx turns it into a Poisson algebra.
  3. For a vertex operator algebra (V,Y,ω,1), the space V / C2(V) is a poission algebra with {a,b} = a0b and a \cdot b =a_{-1}b. For certain vertex operator algebras, these Poisson alegbras are finite dimensional.

See also

Poisson manifold, Poisson superalgebra, antibracket algebra

See also: Poisson algebra, Antibracket algebra, Associative algebra, Bilinear, Derivation, Field (mathematics), Leibniz' law, Lie algebra, Lie bracket