Main Page | See live article | Alphabetical index A Poisson algebra is an associative algebra with a Lie bracket, called the Poisson bracket, satisfying Leibniz' law. More precisely, a Poisson algebra is a vector space over field K equipped with two bilinear products, and [,] such that forms an associative 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 functor). Examples The space of smooth functions over a symplectic manifold. If A is a noncommutative associative algebra, then the commutator [x,y]≡xy-yx turns it into a Poisson algebra. This article is a stub. You can help Wikipedia by fixing it. This article is from Wikipedia. All text is available under the terms of the GNU Free Documentation License.