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Journal of Lie Theory 20 (2010), No. 3, 483--517
Copyright Heldermann Verlag 2010

Local and Global Aspects of Lie Superposition Theorem

David Blázquez-Sanz
Universidad Sergio Arboleda, Escuela de Matemáticas, Calle 74 no. 14-14, Bogotá D.C., Colombia

Juan J. Morales-Ruiz
Universidad Politécnica de Madrid, Escuela Superior de Ingenieros de Caminos, c/ Profesor Aranguren s/n, 28040 Madrid, Spain

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 prove that an ordinary differential equation admit a superposition law if and only if 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.

Keywords: Non-linear superposition laws, Lie-Vessiot systems, Lie-Scheffers theorem, Galois theory of differential equations.

MSC: 34M15, 35C05, 34M35, 34M45

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