Differential Galois Theory and Lie-Vessiot Systems
Analytic and Algebraic Theory of Lie-Vessiot Systemsand Superposition Laws for Ordinary DifferentialEquations
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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 f...
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.
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