Polynomial Poisson structures on affine solvmanifolds

A $n$-dimensional Lie group $G$ equipped with a left invariant symplectic form ${\text{om}}^{+}$ is called a symplectic Lie group. It is well-known that ${\text{om}}^{+}$ induces a left invariant affine structure on $G$. Relatively to this affine structure we show that the left invariant Poisson tensor ${\pi }^{+}$ corresponding to ${\text{om}}^{+}$ 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 ${\text{om}}^{+}$ is also polynomial and the volume form ${\wedge }^{\frac{n}{2}}{\text{om}}^{+}$ 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.