Differential Galois Theory and Lie Symmetries

TL;DR

This paper explores the relationship between differential Galois groups and Lie symmetries in linear differential systems, revealing how symmetries influence the structure of the Galois group through rational solutions and series expansions.

Contribution

It establishes a connection between symmetries and the differential Galois group, introducing a hierarchy of linear systems linked to symmetries and their series expansions.

Findings

Symmetries can be characterized as solutions to a hierarchy of linear differential systems.

Rational symmetries impose specific constraints on the differential Galois group.

The constraints depend on the Maclaurin series of the symmetry at zero.

Abstract

We study the interplay between the differential Galois group and the Lie algebra of infinitesimal symmetries of systems of linear differential equations. We show that some symmetries can be seen as solutions of a hierarchy of linear differential systems. We show that the existence of rational symmetries constrains the differential Galois group in the system in a way that depends of the Maclaurin series of the symmetry along the zero solution.