Definition
A Poisson algebra is a vector space over a field K equipped with two bilinear products, ⋅ and {, }, having the following properties:
- The product ⋅ forms an 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 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.
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