The Poisson algebra reference article from the English Wikipedia on 24-Jul-2004 (provided by Fixed Reference: snapshots of Wikipedia from wikipedia.org)

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, and [,] such that 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.

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