Polynomial Poisson structures on affine solvmanifolds (2011)
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 uni- modular, the symplectic form ω+ is also polynomial and the volume form ∧n/2ω+ 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.