%0 Journal Article
%T Local and Global Aspects of Lie's Superposition Theorem
%A David Bl¨¢zquez-Sanz
%A Juan Jos¨¦ Morales-Ruiz
%J Mathematics
%D 2009
%I arXiv
%X In this paper we give the global conditions for an ordinary differential equation to admit a superposition law of solutions in the classical sense. This completes the well-known Lie superposition theorem. We introduce rigorous notions of pretransitive Lie group action and Lie-Vessiot systems. We proof that pretransitive Lie group actions are transitive. We proof that an ordinary differential equation admit a superposition law if and only if it is a pretransitive Lie-Vessiot system. It means that its enveloping algebra is spanned by fundamental fields of a pretransitive Lie group action. We discuss the relationship of superposition laws with differential Galois theory and review the classical result of Lie.
%U http://arxiv.org/abs/0901.4478v2