The Poisson linearization problem for $\mathfrak{sl}_2(\mathbb{C})$. Part II: The Nash-Moser method

TL;DR

This paper proves a linearization theorem for Poisson structures near zero for $rak{sl}_2(C)$ using a Nash-Moser approach, extending to more general Lie algebras.

Contribution

It develops a Nash-Moser method for flat functions and applies it to linearize Poisson structures, generalizing previous results for $rak{sl}_2(C)$.

Findings

Established linearization of Poisson structures near zero for $rak{sl}_2(C)$

Developed a Nash-Moser technique for flat functions at a point

Extended linearization results to a broader class of Lie algebras

Abstract

This is the second 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 the first part, we calculated the Poisson cohomology associated to $\mathfrak{sl}_2(\mathbb{C})$, and we constructed bounded homotopy operators for the Poisson complex of multivector fields that are flat at the origin. In this second part, we obtain the linearization result, which works for a more general class of Lie algebras. For the proof, we develop a Nash-Moser method for functions that are flat at a point.