On the classification of $3$-dimensional complex hom-Lie algebras

TL;DR

This paper classifies 3-dimensional complex hom-Lie algebras with nilpotent twisting maps, analyzes their degenerations, and offers methods applicable to broader algebraic classification problems.

Contribution

It provides a complete classification of hom-Lie structures with nilpotent twists on 3D complex Lie algebras and explores their degenerations, extending techniques for algebraic structure analysis.

Findings

Classification of hom-Lie structures with nilpotent twists

Analysis of degenerations within the classified family

Methodology applicable to other algebraic structures

Abstract

We classify hom-Lie structures with nilpotent twisting map on $3$-dimensional complex Lie algebras, up to isomorphism, and classify all degenerations in such family. The ideas and techniques presented here can be easily extrapolated to study similar problems in other algebraic structures and provide different perspectives from where one can tackle classical open problems of interest in rigid Lie algebras.