No-go theorems for r-matrices in symplectic geometry

If a triangular Lie algebra acts on a smooth manifold, it induces a Poisson bracket on it. In case this Poisson structure is actually symplectic, we show that this already implies the existence of a flat connection on any vector bundle over the manifold the Lie algebra acts on, in particular the tangent bundle. This implies, among other things, that CPn and higher genus Pretzel surfaces cannot carry symplectic structures that are induced by triangular Lie algebras.