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. 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.