Linear Poisson structures on R^4

TL;DR

This paper classifies all 4-dimensional linear Poisson structures related to extensions of 3-dimensional unimodular Lie algebras, providing a complete understanding of their structure and affine Poisson structures on R^3.

Contribution

It offers a comprehensive classification of 4D linear Poisson structures derived from 3D unimodular Lie algebra extensions, filling a gap in Lie algebra and Poisson geometry literature.

Findings

Complete classification of 4D linear Poisson structures

Total classification of affine Poisson structures on R^3

Identification of Lie algebra extensions involved

Abstract

We classify all of the 4-dimensional linear Poisson structures of which the corresponding Lie algebras can be considered as the extension by a derivation of 3-dimensional unimodular Lie algebras. The affine Poisson structures on R^3 are totally classified.