Simplicity in the Faulkner construction

TL;DR

This paper revisits the Faulkner construction for metric 3-Leibniz algebras with positive-definite signature, clarifying the notion of simplicity and linking it to existing classifications of simple Lie (super)algebras.

Contribution

It establishes a relationship between simplicity notions of 3-Leibniz algebras, their representations, and embedding Lie (super)algebras, simplifying their classification.

Findings

Relates simplicity of 3-Leibniz algebras to simple Lie (super)algebras

Reduces classification problem to known classifications of simple Lie (super)algebras

Clarifies notions of simplicity in the context of the Faulkner construction

Abstract

We revisit the Faulkner construction of metric 3-Leibniz algebras admitting an embedding Lie (super)algebra. In the case of positive-definite signature, we relate the various notions of simplicity: of the 3-algebra, of the representation and of the embedding Lie (super)algebra. This reduces their classification to the extant classifications of simple Lie (super)algebras.