Poisson algebra
From Exampleproblems
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).
Poisson algebras are employed in Hamiltonian mechanics.
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.
- For a vertex operator algebra (V,Y,ω,1), the space V / C2(V) is a Poisson algebra with {a,b} = a0b and
. For certain vertex operator algebras, these Poisson algebras are finite dimensional.
