Algebras of Complete Hörmander Vector Fields, and Lie-Group Construction

The aim of this note is to characterize the Lie algebras g of the analytic vector fields in RN which coincide with the Lie algebras of the (analytic) Lie groups defined on RN (with its usual differentiable structure). We show that such a characterization amounts to asking that: (i) g is N-dimensional; (ii) g admits a set of Lie generators which are complete vector fields; (iii) g satisfies Hörmander’s rank condition. These conditions are necessary, sufficient and mutually independent. Our approach is constructive, in that for any such g we show how to construct a Lie group G = (RN, *) whose Lie algebra is g. We do not make use of Lie’s Third Theorem, but we only exploit the Campbell-Baker-Hausdorff-Dynkin Theorem for ODE’s.