%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