No-go theorems for $ r $-matrices in symplectic geometry

Abstract

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 $ \mathbb{C}P^n $ and higher genus Pretzel surfaces cannot carry symplectic structures that are induced by triangular Lie algebras.