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

TL;DR

This paper proves a linearization theorem for Poisson structures related to rak{sl}_2(\u00a3) by computing Poisson cohomology and constructing homotopy operators, laying groundwork for a broader linearization result in the second part.

Contribution

It calculates Poisson cohomology for rak{sl}_2(a3) and develops tools for linearizing Poisson structures, advancing understanding of their local behavior.

Findings

Poisson cohomology for rak{sl}_2(a3) computed

Bounded homotopy operators constructed for Poisson complex

Linearization theorem proved for structures with rak{sl}_2(a3) linear approximation

Abstract

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.