Lie's reduction method and differential Galois theory in the complex analytic context

Abstract

This paper is dedicated to the differential Galois theory in the complex analytic context for Lie-Vessiot systems. Those are the natural generalization of linear systems, and the more general class of differential equations adimitting superposition laws, as recently stated in [5]. A Lie-Vessiot system is automatically translated into a equation in a Lie group that we call automorphic system. Reciprocally an automorphic system induces a hierarchy of Lie-Vessiot systems. In this work we study the global analytic aspects of a classical method of reduction of differential equations, due to S. Lie. We propose an differential Galois theory for automorphic systems, and explore the relationship between integrability in terms of Galois theory and the Lie's reduction method. Finally we explore the algebra of Lie symmetries of a general automorphic system.