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

Abstract

If [Formula: see text] is a semisimple Lie algebra of vector fields on [Formula: see text] with a split Cartan subalgebra [Formula: see text], then it is proved here that the dimension of the generic orbit of [Formula: see text] coincides with the dimension of [Formula: see text]. As a consequence one obtains a local canonical form of [Formula: see text] in terms of exponentials of coordinate functions and vector fields that are independent of these coordinates — for a suitable choice of coordinate system. This result is used to classify semisimple algebras of local vector fields on [Formula: see text] and to determine all representations of [Formula: see text] as local vector fields on [Formula: see text]. These representations are in turn used to find linearizing coordinates for any second-order ordinary differential equation that admits [Formula: see text] as its symmetry algebra and for a system of two second-order ordinary differential equations that admits [Formula: see text] as its symmetry algebra.