A global lifting for finite-dimensional lie algebras of complete vector fields

Motivated by the so-called Lifting and Approximation Theorem by Rothschild and Stein, we consider a set of vector fields Formula: see text on a manifold Formula: see text, and we study the problem of obtaining a global lifting of the Formula: see text’s to a system of generators of the Lie algebra Formula: see text of a Lie group Formula: see text. By assuming that the Lie algebra Formula: see text generated by Formula: see text is finite-dimensional and all of the Formula: see text’s are complete vector fields, but without the assumption that they satisfy Hörmander’s rank condition, we reduce the lifting problem to a result of Palais on integrability. This proves that any Formula: see text in Formula: see text is Formula: see text-related to a left invariant vector field Formula: see text, where Formula: see text is a smooth map resulting from a right action Formula: see text of Formula: see text on Formula: see text. Both the lifting map Formula: see text and the lifting vector fields Formula: see text are globally defined, and our result generalizes the global Lifting obtained by Folland in the special case of dilation-invariant vector fields. According to Palais’ Integrability, the map Formula: see text is obtained via the flow of suitable vector fields on Formula: see text and the image set of Formula: see text, namely, Formula: see text, is the Sussmann orbit of Formula: see text through Formula: see text. Examples are provided, showing that a germ of Formula: see text, obtained through the integration of Formula: see text and depending only on the Baker–Campbell–Hausdorff formula, is often sufficient to get the global lifting without the need of abstract results.