Differential Galois Theory and Lie-Vessiot Systems

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.