Projectable Lie algebras of vector fields in 3D

TL;DR

This paper classifies and constructs Lie algebras of vector fields in three dimensions by lifting from two-dimensional cases, identifying three main types of transitive lifts and their cohomological properties.

Contribution

It extends Lie's classification to three dimensions by systematically constructing and analyzing all possible transitive lifts of 2D Lie algebras, revealing their cohomological structure.

Findings

Identified three main types of transitive lifts in 3D

Computed all lifts for Lie's 2D classification

Connected lifts to Lie algebra cohomology

Abstract

Starting with Lie's classification of finite-dimensional transitive Lie algebras of vector fields on $\mathbb C^2$ we construct Lie algebras of vector fields on the bundle $\mathbb C^2 \times \mathbb C$ by lifting the Lie algebras from the base. There are essentially three types of transitive lifts and we compute all of them for the Lie algebras from Lie's classification. The simplest type of lift is encoded by Lie algebra cohomology.