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. Summary of resultsThroughout the first section we review the concept of Lie-Vessiot system, and we state the global version of Lie's superposition theorem; it is a recallment of results contained in previous works [5] and [4]. Second section is devoted to the notion of automorphic system. We study its geometry and the relationship with general Lie-Vessiot systems: it is introduced the concept of Lie-Vessiot hierarchy, that relates an automorphic system with a family of Lie-Vessiot systems induced in homogeneous spaces. In the third section we introduce Lie's reduction method (Theorem 3.5) in the complex analytic context. We note that Lie's reduction method is local in the time parameter, and then we explore the obstructions to the existence of a reduction method global in time (Proposition 3.9). In the fourth section we propose a Galois theory for automorphic systems, based on the Lie reduction method. In our theory, the Galois group is the smallest group

Lie systems: theory, generalisations, and applications

“…As a result of the analysis of the former themes, Lie systems provide methods to study the integrability of systems of first-order differential equations [40], Control Theory [32,61,79,187], geometric phases [98], certain problems in Quantum Mechanics [46,51], and other topics. Finally, it is remarkable that the theory of Lie systems has been investigated by means of different techniques and approaches, like Galois theory [17,19] or Differential Geometry [38,60,186,220].…”

Differential Galois Theory and Lie Symmetries

“…Here we recall some of the definitions from [5], adapted to the particular case of linear equations. Let E be a finite-dimensional complex vector space and (E, Ψ) a linear representation of GL(n, C) in E. The group morphism Ψ induces a Lie algebra morphism Ψ :…”

“…In order to link Lie symmetries with differential Galois theory, we follow a geometrical approach developed by the first two authors in [4]. Some general results about symmetries were already stated in [5,Section 6], but in a more general context of automorphic systems. In particular, it is implicit in [5] that the eigenring [3,20] consists of vertical Lie symmetries.…”

Multisymplectic structures and invariant tensors for Lie systems

“…For instance, since symplectic structures are multisymplectic ones, the Lie-Hamilton systems related to symplectic structures [24] can be considered as multisymplectic Lie systems. Moreover, we prove that the hereafter called locally automorphic Lie systems, which are Lie systems locally diffeomorphic to the very relevant automorphic Lie systems [10,20,42], can always be studied through multisymplectic Lie systems (cf. Theorem 4.9).…”