Three-dimensional homogeneous spaces with non-solvable transformation groups

TL;DR

This paper classifies all transitive Lie algebra actions on three-dimensional spaces, highlighting the differences between solvable and non-solvable cases using invariant foliation structures.

Contribution

It provides a comprehensive classification of transitive Lie algebra actions on C^3 and R^3, emphasizing the limitations in extending results to solvable Lie algebras.

Findings

Complete classification of non-solvable transitive Lie algebra actions on three-dimensional spaces.

Identification of structural differences between solvable and non-solvable cases.

Use of invariant foliation structures as a key technical tool.

Abstract

We classify all transitive actions of Lie algebras of vector fields on C^3 and R^3 up to a local equivalence and discuss why this classification can not be extended in general to the solvable case. The main technical tool is the structure of one-dimensional invariant foliations on homogeneous spaces.