The Poisson linearization problem for $\mathfrak{sl}_2(\mathbb{C})$. Part I: Poisson cohomology

This is the first of two papers, in which we prove a version of Conn's linearization theorem for the Lie algebra $\mathfrak{sl}_2(\mathbb{C})\simeq \mathfrak{so}(3,1)$. Namely, we show that any Poisson structure whose linear approximation at a zero is isomorphic to the Poisson structure associated to $\mathfrak{sl}_2(\mathbb{C})$ is linearizable. In this first part, we calculate the Poisson cohomology associated to $\mathfrak{sl}_2(\mathbb{C})$, and we construct bounded homotopy operators for the Poisson complex of multivector fields that are flat at the origin. In the second part, we will obtain the linearization result, which works for a more general class of Lie algebras. For the proof, we will develop a Nash-Moser method for functions that are flat at a point.