
Journal of Lie Theory 20 (2010), No. 3, 483517 Copyright Heldermann Verlag 2010 Local and Global Aspects of Lie Superposition Theorem David BlázquezSanz Universidad Sergio Arboleda, Escuela de Matemáticas, Calle 74 no. 1414, Bogotá D.C., Colombia david.blazquezsanz@usa.edu.co Juan J. MoralesRuiz Universidad Politécnica de Madrid, Escuela Superior de Ingenieros de Caminos, c/ Profesor Aranguren s/n, 28040 Madrid, Spain juan.moralesruiz@upm.es We give the global conditions for an ordinary differential equation to admit a superposition law of solutions in the classical sense. This completes the wellknown Lie superposition theorem. We introduce rigorous notions of pretransitive Lie group action and LieVessiot 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: Nonlinear superposition laws, LieVessiot systems, LieScheffers theorem, Galois theory of differential equations. MSC: 34M15, 35C05, 34M35, 34M45 [ Fulltextpdf (312 KB)] for subscribers only. 