The Poisson linearization problem for $\mathfrak{sl}_2(\mathbb{C})$. Part II: The Nash-Moser method
Year: 2022
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Marcut, Ioan, Zeiser, Florian
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4311723948