Automorphisms of Real 4 Dimensional Lie Algebras and the Invariant Characterization of Homogeneous 4-Spaces

TL;DR

This paper comprehensively classifies automorphisms of all 4-dimensional real Lie algebras and uses their action to characterize homogeneous 4-spaces invariantly, aiding geometric and algebraic understanding.

Contribution

It provides a complete description of automorphisms of 4D real Lie algebras and introduces an invariant method to classify homogeneous 4-spaces.

Findings

Complete automorphism classification for all 4D real Lie algebras

Invariant characterization of 4-dimensional homogeneous spaces

Action on symmetric positive definite matrices defines equivalence classes

Abstract

The automorphisms of all 4-dimensional, real Lie Algebras are presented in a comprehensive way. Their action on the space of $4\times 4$, real, symmetric and positive definite, matrices, defines equivalence classes which are used for the invariant characterization of the 4-dimensional homogeneous spaces which possess an invariant basis.