Transitive Lie algebras of vector fields---an overview

TL;DR

This paper provides a concise overview of transitive Lie algebras of vector fields, highlighting their historical development, key theories, and significance in symmetry analysis of differential equations.

Contribution

It offers a subjective, accessible summary of the theory of transitive and primitive Lie algebras of vector fields, emphasizing foundational work by Lie and others.

Findings

Summarizes the theory of transitive Lie algebras of vector fields.

Highlights historical development and key contributors.

Provides an accessible overview without exhaustive detail.

Abstract

This overview paper is intended as a quick introduction to Lie algebras of vector fields. Originally introduced in the late 19th century by Sophus Lie to capture symmetries of ordinary differential equations, these algebras, or infinitesimal groups, are a recurring theme in 20th-century research on Lie algebras. I will focus on so-called transitive or even primitive Lie algebras, and explain their theory due to Lie, Morozov, Dynkin, Guillemin, Sternberg, Blattner, and others. This paper gives just one, subjective overview of the subject, without trying to be exhaustive.