Lie's Reduction Method and Differential Galois Theory in the Complex Analytic Context

TL;DR

This paper develops a differential Galois theory for Lie-Vessiot systems in the complex analytic setting, linking Lie's reduction method with integrability and symmetry analysis of automorphic systems.

Contribution

It introduces a new Galois theory for automorphic systems and connects Lie's reduction method with integrability and symmetry structures.

Findings

Established a differential Galois theory for automorphic systems.

Linked Lie's reduction method with integrability criteria.

Analyzed the algebra of Lie symmetries in automorphic systems.

Abstract

This paper is dedicated to the differential Galois theory in the complex analytic context for Lie-Vessiot systems. Those are the natural generaliza- tion of linear systems, and the more general class of differential equations adimitting superposition laws, as recently stated in [5]. A Lie-Vessiot sys- tem 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 the- ory and the Lie's reduction method. Finally we explore the algebra of Lie symmetries of a general automorphic system.