Deformations of the Lie-Poisson sphere of a compact semisimple Lie algebra

TL;DR

This paper explicitly computes the moduli space of Poisson structures on the sphere associated with a compact semisimple Lie algebra, revealing new insights into deformations of degenerate Poisson structures in higher dimensions.

Contribution

It provides the first explicit computation of a Poisson moduli space in dimension three or higher around a degenerate structure.

Findings

Computed the moduli space of Poisson structures on the sphere

Identified the structure of deformations around a degenerate Poisson structure

Extended understanding of Poisson geometry in higher dimensions

Abstract

A compact semisimple Lie algebra $\mathfrak{g}$ induces a Poisson structure $\pi$ on the unit sphere $S$ in $\mathfrak{g}^*$. We compute the moduli space of Poisson structures on $S$ around $\pi$. This is the first explicit computation of a Poisson moduli space in dimension greater or equal than three around a degenerate (i.e. not symplectic) Poisson structure.