Local and Global Aspects of Lie's Superposition Theorem

TL;DR

This paper extends Lie's superposition theorem by establishing global conditions for differential equations to admit superposition laws, introducing new concepts like pretransitive Lie group actions and Lie-Vessiot systems.

Contribution

It provides a complete characterization of when an ODE admits a superposition law using global group action conditions, completing Lie's classical theorem.

Findings

Pretransitive Lie group actions are transitive.

An ODE admits a superposition law iff it is a pretransitive Lie-Vessiot system.

The enveloping algebra is spanned by fundamental fields of a pretransitive Lie group action.

Abstract

In this paper we give the global conditions for an ordinary differential equation to admit a superposition law of solutions in the classical sense. This completes the well-known Lie superposition theorem. We introduce rigorous notions of pretransitive Lie group action and Lie-Vessiot systems. We proof that pretransitive Lie group actions are transitive. We proof that an ordinary differential equation admit a superposition law if and only if it is a pretransitive Lie-Vessiot system. It means that its enveloping algebra is spanned by fundamental fields of a pretransitive Lie group action. We discuss the relationship of superposition laws with differential Galois theory and review the classical result of Lie.