Editing Poisson algebra (section)

==Definition==
A Poisson algebra is a [[vector space]] over a [[field (mathematics)|field]] ''K'' equipped with two [[bilinear map|bilinear]] products, ⋅ and {, }, having the following properties:

* The product ⋅ forms an  [[associative algebra|associative ''K''-algebra]].
* The product {, }, called the [[Poisson bracket]], forms a [[Lie algebra]], and so it is anti-symmetric, and obeys the [[Jacobi identity]].
* The Poisson bracket acts as a [[Derivation (abstract algebra)|derivation]] of the associative product ⋅, so that for any three elements ''x'', ''y'' and ''z'' in the algebra, one has {''x'', ''y'' ⋅ ''z''} = {''x'', ''y''} ⋅ ''z'' + ''y'' ⋅ {''x'', ''z''}.

The last property often allows a variety of different formulations of the algebra to be given, as noted in the examples below.