On finite dimensional Lie algebras of planar vector fields with rational coefficients

TL;DR

This paper classifies all finite-dimensional Lie algebras that can be embedded as subalgebras within the Lie algebra of planar vector fields with rational function coefficients over an algebraically closed field of characteristic zero.

Contribution

It provides a complete classification of finite-dimensional subalgebras of the Lie algebra of planar rational vector fields, a problem previously unresolved.

Findings

Identified all possible finite-dimensional subalgebras

Characterized the structure of these subalgebras

Provided explicit realizations within the algebra

Abstract

The Lie algebra of planar vector fields with coefficients from the field of rational functions over an algebraically closed field of characteristic zero is considered. We find all finite-dimensional Lie algebras that can be realized as subalgebras of this algebra.