Poisson algebra

Definition

A Poisson algebra is an associative algebra $A$ with a Poisson bracket.

Example

Probably the most important example will be the associative algebra of smooth functions $C^{\infty }(M)$ for a symplectic manifold $(M,\omega )$. The Poisson bracket is defined on smooth functions $f,g:M\to \mathbb{R}$ as $\{f,g\}=\omega (X_f,X_g)$ where $X_f,X_g$ are the Hamiltonian vector fields of the functions $f$ and $g$.