Realizations of Real Low-Dimensional Lie Algebras

TL;DR

This paper introduces a novel technique using megaideals to classify all inequivalent realizations of real Lie algebras up to dimension four in vector fields, extending previous classifications to arbitrary variable spaces.

Contribution

It presents a comprehensive classification method for low-dimensional real Lie algebras using megaideals, generalizing earlier results to higher-dimensional variable spaces.

Findings

Complete set of inequivalent realizations for Lie algebras up to dimension four

Generalization of classification results to arbitrary variable spaces

Enhanced understanding of automorphisms and subalgebra structures

Abstract

Using a new powerful technique based on the notion of megaideal, we construct a complete set of inequivalent realizations of real Lie algebras of dimension no greater than four in vector fields on a space of an arbitrary (finite) number of variables. Our classification amends and essentially generalizes earlier works on the subject. Known results on classification of low-dimensional real Lie algebras, their automorphisms, differentiations, ideals, subalgebras and realizations are reviewed.