Classification problem of simple Hom-Lie algebras

TL;DR

This paper investigates the classification of simple Hom-Lie algebras, introduces strongly simple Hom-Lie algebras, and provides a complete classification of regular simple cases over arbitrary fields.

Contribution

It introduces the concept of strongly simple Hom-Lie algebras and classifies regular simple Hom-Lie algebras over any field, clarifying their structure and isomorphism classes.

Findings

Complete classification of regular simple Hom-Lie algebras over any field.

All simple 3-dimensional anticommutative algebras are Yau twists of so(3).

All 2-dimensional simple Hom-Lie algebras are classified, correcting previous misconceptions.

Abstract

First, we construct some families of nonsolvable anticommutative algebras, solvable Lie algebras and even nilpotent Lie algebras, that can be endowed with the structure of a simple Hom-Lie algebra. This situation shows that a classification of simple Hom-Lie algebras would be unrealistic without any further restrictions. We introduce the class of \emph{strongly simple Hom-Lie algebras}, as the class of anticommutative algebras that are simple Hom-Lie with respect to all their twisting maps. We show some of its properties, provide a characterization and explore some of its subclasses. Furthermore, we provide a complete classification of regular simple Hom-Lie algebras over any arbitrary field, together with a description of a lower bound of the number of their isomorphism classes, which depends entirely on the finiteness or not of the underlying field. In addition, we establish that every simple anticommutative algebra of dimension $3$ turns out to be the outside Yau's twist of the special orthogonal Lie algebra $\mathfrak{so}(3,\mathbb{F})$ with respect to some bijective linear map. Also, we determine all the simple Hom-Lie algebras of dimension $2$, up to conjugacy, which were wrongly claimed to be nonexistent in previous literature. Finally, we establish a new \emph{simplicity criterion} for Lie algebras, which as an application shows that the simplicity in the category of multiplicative Hom-Lie algebras is equivalent to that of anticommutative algebras.