Poisson ring

Definition
A Poisson ring is the following data:


 * 1) An abelian group $$A$$.
 * 2) An associative ring structure on $$A$$ with the given underlying abelian group, and with multiplication $$*$$
 * 3) A Lie ring structure on $$A$$, with the same underlying abelian group, and with Lie bracket denoted $$\{, \}$$ -- this is called the Poisson bracket.

satisfying the following compatibility condition -- the Poisson bracket on the left acts as derivations of the associative ring, i.e.:

$$\{ a, b * c \} = \{ a,b \} * c + b * \{ a,c \} \ \forall a,b,c \in A$$

Facts

 * Associated Poisson ring of an associative ring: For any associative ring $$A$$, there is a natural Poisson ring structure where the Poisson bracket is the additive commutator (see also associated Lie ring of an associative ring).