# 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, [itex]\cdot[itex] and [,] such that [itex]\cdot[itex] 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]≡xyyx turns it into a Poisson algebra.
3. For a vertex operator algebra [itex](V,Y, \omega, 1)[itex], the space [itex]V/C_2(V)[itex] is a poission algebra with [itex]\{a,b\}=a_0b[itex] and [itex]a \cdot b =a_{-1}b[itex]. For certain vertex operator algebras, these Poisson alegbras are finite dimensional.

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