§15§The purpose of this work is to develop a differentialGalois theory for differential equations admittingsuperposition laws. First, we characterize thosedifferential equations in terms of Lie group actions,generalizing some classical results due to S. Lie. Wecall them Lie-Vessiot systems. Then, we develop adifferential Galois theory for Lie-Vessiot systemsboth in the complex analytic and algebraic contexts.In the complex analytic context we give a theory thatgeneralizes the tannakian approach to the classicalPicard-Vessiot theory. In the algebraic case, westudy differential equations under the formalism ofdifferential algebra. We prove that algebraicLie-Vessiot systems are solvable in strongly normalextensions. Therefore, Lie-Vessiot systems aredifferential equations attached to the Kolchin'sdifferential Galois theory. §16§The purpose of this work is to develop a differential§Galois theory for differential equations admitting§superposition laws. First, we characterize those§differential equations in terms of Lie group actions,§generalizing some classical results due to S. Lie. We§call them Lie-Vessiot systems. Then, we develop a§differential Galois theory for Lie-Vessiot systems§both in the complex analytic and algebraic contexts.§In the complex analytic context we give a theory that§generalizes the tannakian approach to the classical§Picard-Vessiot theory. In the algebraic case, we§study differential equations under the formalism of§differential algebra. We prove that algebraic§Lie-Vessiot systems are solvable in strongly normal§extensions. Therefore, Lie-Vessiot systems are§differential equations attached to the Kolchin''s§differential Galois theory. §01§Blázquez-Sanz David David Blázquez Sanz is a Spanish-American mathematician. Heobtained his degree in Universidad de Salamanca and his doctoratein Universitat Politècnica de Catalunya, under the supervision ofJuan J. Morales Ruiz. Nowadays he is professor in Sergio ArboledaUniversity, in Bogotá, Colombia. He also studies and teacheschinese martial arts.
