Automorphisms of real Lie algebras of dimension five or less

TL;DR

This paper presents a method based on the Lie algebra Krull-Schmidt Theorem to classify automorphisms of all indecomposable real Lie algebras of dimension five or less, simplifying the automorphism group classification process.

Contribution

It formulates and proves a Lie algebra version of the Krull-Schmidt Theorem, enabling systematic construction of automorphisms from indecomposable components.

Findings

Classified automorphisms of all indecomposable real Lie algebras of dimension ≤ 5.

Provided a concise tabular presentation of results.

Simplified automorphism classification for finite-dimensional Lie algebras.

Abstract

The Lie algebra version of the Krull-Schmidt Theorem is formulated and proved. This leads to a method for constructing the automorphisms of a direct sum of Lie algebras from the automorphisms of its indecomposable components. For finite-dimensional Lie algebras, there is a well-known algorithm for finding such components, so the theorem considerably simplifies the problem of classifying the automorphism groups. We illustrate this by classifying the automorphisms of all indecomposable real Lie algebras of dimension five or less. Our results are presented very concisely, in tabular form.