Lie $3-$algebra and super-affinization of split-octonions

TL;DR

This paper extends the concept of Lie 3-algebras to split-octonions, unifies their product, verifies algebraic properties, and constructs a super-symmetric affine super-algebra with potential applications in gauge theory.

Contribution

It introduces a unified product for split-octonions, verifies their Malcev algebra structure, and constructs a super-symmetric affine super-algebra based on split-octonions.

Findings

$ ext{SO}$ is a Malcev algebra

Recalculated structure constants for $ ext{SO}$

Constructed $ ext{N}=1$ super-symmetric $ ext{SO}$ affine super-algebra

Abstract

The purpose of this study is to extend the concept of a generalized Lie $3-$ algebra, known to the divisional algebra of the octonions $\mathbb{O}$, to split-octonions $\mathbb{SO}$, which is non-divisional. This is achieved through the unification of the product of both of the algebras in a single operation. Accordingly, a notational device is introduced to unify the product of both algebras. We verify that $\mathbb{SO}$ is a Malcev algebra and we recalculate known relations for the structure constants in terms of the introduced structure tensor. Finally we construct the manifestly super-symmetric $\mathcal{N}=1\;\mathbb{SO}$ affine super-algebra. An application of the split Lie $3-$algebra for a Bagger and Lambert gauge theory is also discussed.