Whar are the differences between Poisson algebras with associative and Lie algebras?

Question

Definition of Poisson algebras can be found here. The tensor algebra of a Lie algebra has a Poisson algebra structure. Moreover, by imposing an associative algebra a commutator, it turns it into Poisson algebra. These are considered as two examples of Poisson algebras. Now I am confused of definition of Poisson algebras in the sense of its difference with associative and Lie algebras? In the Poisson bracket $\{ , \}$ which is actually written as $\{a,b\}=a b - b a$, is $ab$ equivalent to $a \cdot b$, where $\cdot$ is the bilinear product from associative algebra in the definition of Poisson algebras?

Answer 1

Every associative algebra $A$ has a canonical Poisson algebra structure where the bracket is given by commutator $\{ a, b \} = ab - ba$, as you say. This is not the only possible Poisson bracket on $A$. This particular bracket vanishes iff $A$ is commutative, but commutative algebras can have other interesting (and in particular nonvanishing) Poisson brackets on them (and so can associative algebras).

For example, $\mathbb{R}[x, y]$ has a Poisson bracket satisfying $\{ x, y \} = 1$ (all other brackets are determined by the Leibniz rule); this is the Poisson algebra of polynomial functions on the cotangent bundle $T^{\ast}(\mathbb{R})$, which is even a symplectic manifold. Also, if $\mathfrak{g}$ is any Lie algebra, the symmetric algebra $S(\mathfrak{g})$ has a Poisson bracket extending the Lie bracket on $\mathfrak{g}$; this is the Poisson algebra of polynomial functions on the Poisson manifold $\mathfrak{g}^{\ast}$.