Polynomial Poisson structures on affine solvmanifolds

TL;DR

This paper demonstrates that on affine solvmanifolds with symplectic structures, invariant Poisson tensors are polynomial of degree one, and extends this polynomiality property to various invariant tensor fields, especially on nilpotent groups.

Contribution

It establishes the polynomial nature of invariant Poisson and tensor fields on affine symplectic Lie groups and solvmanifolds, including nilpotent cases, with implications for compact quotients.

Findings

Invariant Poisson tensors are degree 1 polynomials on affine symplectic Lie groups.

Left invariant tensor fields on nilpotent symplectic Lie groups are polynomial.

Many symplectic Lie groups admit polynomial Poisson structures on compact quotients.

Abstract

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.