Equality of the algebraic and geometric ranks of Cartan subalgebras and applications to linearization of a system of ordinary differential equations

TL;DR

This paper proves the equality of algebraic and geometric ranks of Cartan subalgebras in semisimple Lie algebras of vector fields, leading to classification results and methods for linearizing certain differential equations.

Contribution

It establishes the equality of algebraic and geometric ranks of Cartan subalgebras and applies this to classify vector field algebras and linearize specific ODE systems.

Findings

Dimension of generic orbit of Cartan subalgebra equals its rank

Classification of semisimple vector field algebras on R^3

Explicit linearization methods for certain ODEs with symmetry algebra sl(3,R) or sl(4,R)

Abstract

If $L$ is a semisimple Lie algebra of vector fields on R^N with a split Cartan subalgebra C, then it is proved that the dimension of the generic orbit of C coincides with the dimension of C. As a consequence one obtains a local canonical form of L in terms of exponentials of coordinate functions and vector fields that are independent of these coordinates -- for a suitable choice of coordinates. This result is used to classify semisimple algebras of vector fields on R^3 and to determine all representations of sl(N, R) as vector fields on R^N. These representations are used to find linearizing coordinates for any second order ordinary differential equation that admits sl(3, R) as its symmetry algebra and for a system of two second order ordinary differential equations that admits sl(4, R) as its symmetry algebra.