A n-dimensional Lie group G equipped with a left invariant symplectic form ω + is called a symplectic Lie group.It is well known that ω + induces a left invariant affine structure on G. Relative to this affine structure we show that the left invariant Poisson tensor π + corresponding to ω + is polynomial of degree at most 1 and any right invariant k-multivector field on G is polynomial of degree at most k.If G is unimodular, the symplectic form ω + is also polynomial and the volume form ∧ n/2 ω + 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.