From Lie Algebras to Lie Groups within Synthetic Differential Geometry:Weil Sprouts of Lie's Third Fundamental Theorem

TL;DR

This paper demonstrates how to construct Weil prolongations of Lie groups from given Lie algebras within synthetic differential geometry, addressing a longstanding problem in infinite-dimensional Lie group theory.

Contribution

It introduces a method to construct Weil prolongations of hypothetical Lie groups from Lie algebras using synthetic differential geometry, advancing understanding of Lie group-Lie algebra correspondence.

Findings

Constructed Weil prolongations from Lie algebras

Addressed the existence problem of Lie groups with given Lie algebras

Applied synthetic differential geometry to Lie theory

Abstract

Weil prolongations of a Lie group are naturally Lie groups. It is not known in the theory of infinite-dimensional Lie groups how to construct a Lie group with a given Lie algebra as its Lie algebra or whether there exists such a Lie group at all. We will show in this paper how to construct some Weil prolongations of this mythical Lie group from a given Lie algebra. We will do so within our favorite framework of synthetic differential geometry.