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.
@article{4707,
title = {Algebras of Complete H\"ormander Vector Fields, and Lie-Group Construction},
journal = {Bruno Pini Mathematical Analysis Seminar},
year = {2014},
doi = {10.6092/issn.2240-2829/4707},
language = {IT},
url = {http://dml.mathdoc.fr/item/4707}
}
Bonfiglioli, Andrea. Algebras of Complete Hörmander Vector Fields, and Lie-Group Construction. Bruno Pini Mathematical Analysis Seminar, (2014), . doi : 10.6092/issn.2240-2829/4707. http://gdmltest.u-ga.fr/item/4707/