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

2025/07/30 by Hassan Azad, Azad, Hassan, Indranil Biswas +5
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Representation Theory (math.RT) #math.DG #math.RT

paper · pdf · doi:10.48550/arxiv.2507.22642

Final version; to appear in the International Journal of Mathematics

arxiv created 2026/07/30 · arxiv updated 2026/07/31

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 \citeABF2 and \citeABF3 completes the classification of finite dimensional subalgebras of vector fields on the complex plane.

Citations

• Semisimple algebras of vector fields on C^N of maximal rank
• On Lie's classification of subalgebras of vector fields on the plane
• Equality of the algebraic and geometric ranks of Cartan subalgebras and applications to linearization of a system of ordinary differential equations

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