Poisson cohomology and linearization for simple Lie algebras

forthcoming

Given a geometric structure, can we find local coordinates such that the structure has a particular nice expression? That is one of the most fundamental questions for any geometric structure, that of a local normal form. In Poisson geometry, this question is directly related to the study of Lie algebras, as any Lie algebra is equivalent to a linear Poisson structure on its dual. Given a Poisson structure with a singularity, can we locally identify the Poisson structure always with its linear (first order) version around the singularity? This question was first asked by Weinstein his seminal work on Poisson manifolds in 1983. However, even for (semi)simple Lie algebras, this question has not been completely resolved, despite several results. In particular, it is not know for (semi)simple Lie algebra of real rank one with semisimple compact part. In this project we aim to resolve the problem for all remaining simple Lie algebras, taking an important step towards a complete answer. In the lowest dimensional case, that of so(3,1), an affirmative, positive answer has been provided by myself in my PhD thesis. This is the first non-compact example with a known positive answer. The general strategy is the following: 1) Show that the cohomology group controlling the deformation problem vanishes and find sufficiently nice cochain homotopies 2) Apply a Nash-Moser type inverse function theorem to establish linearization Generalizing the ideas of the proof to all simple Lie algebras will bring together techniques from foliation theory, symplectic geometry, homological algebra. Additionally it will require developing and applying new techniques in algebraic geometry, representation theory and functional analysis, specifically adapted to the Lie algebras under consideration and their stratification by (co)adjoint orbits. As such, the result will be highly influential and interesting for several areas of Mathematics.