On Lie's classification of nonsolvable subalgebras of vector fields on the plane

TL;DR

This paper provides a concise proof of Lie's classification of certain finite-dimensional subalgebras of vector fields on the complex plane, utilizing representation theory of sl(2, C), completing the classification effort.

Contribution

It offers a simplified proof of Lie's classification for subalgebras with Levi decomposition, integrating previous results to finalize the classification.

Findings

Complete classification of finite-dimensional subalgebras of vector fields on the complex plane.

Simplified proof leveraging representation theory of sl(2, C).

Integration of prior work to finalize the classification.

Abstract

A brief proof of Lie's classification of finite dimensional subalgebras of vector fields on the complex plane that have a proper Levi decomposition is given. The proof uses basic representation theory of sl(2, C). This, combined with \cite{ABF2} and \cite{ABF3} completes the classification of finite dimensional subalgebras of vector fields on the complex plane.