Poisson algebras appear naturally in Hamiltonian mechanics, and are also central in the study of quantum groups.
The algebra is named in honour of Siméon Denis Poisson.
A Poisson algebra is a vector space over a field K equipped with two bilinear products, ⋅ and {, }, having the following properties: The last property often allows a variety of different formulations of the algebra to be given, as noted in the examples below.
The space of real-valued smooth functions over a symplectic manifold forms a Poisson algebra.
A very explicit construction of this is given in the article on universal enveloping algebras.
Note that the resulting AL should not be confused with the tensor algebra construction described in the previous section.
If one wished, one could also apply that construction as well, but that would give a different Poisson algebra, one that would be much larger.
Gerstenhaber algebras conventionally occur in BRST quantization.