  A $n$-dimensional Lie group $G$ equipped with a left invariant symplectic
form $\om^+$ is called a symplectic Lie group. It is well-known that $\om^+$
induces a left invariant affine structure on $G$. Relatively to this affine
structure we show that the left invariant Poisson tensor $\pi^+$ corresponding
to $\om^+$ is polynomial of degree 1 and any right invariant $k$-multivector
field on $G$ is polynomial of degree at most $k$. If $G$ is unimodular, the
symplectic form $\om^+$ is also polynomial and the volume form
$\wedge^{\frac{n}2}\om^+$ is parallel. We show also that any left invariant
tensor field on a nilpotent symplectic Lie group is polynomial, in particular,
any left invariant Poisson structure on a nilpotent symplectic Lie group is
polynomial. Because many symplectic Lie groups admit uniform lattices, we get a
large class of polynomial Poisson structures on compact affine solvmanifolds.
