J-ternary algebras, structurable algebras, and Lie superalgebras

TL;DR

This paper constructs a new class of Lie superalgebras from J-ternary and structurable algebras in characteristic 3, revealing connections to exceptional simple Lie algebras and introducing a novel magic square of superalgebras.

Contribution

It introduces a method to derive Lie superalgebras from J-ternary algebras, including a new magic square, expanding understanding of algebraic structures in characteristic 3.

Findings

Constructed Lie superalgebras from J-ternary algebras in characteristic 3

Connected some exceptional simple Lie algebras to J-ternary algebras

Developed a new magic square of Lie superalgebras

Abstract

A Lie superalgebra is attached to any finite-dimensional J-ternary algebra over an algebraically closed field of characteristic 3, using a process of semisimplification via tensor categories. Some of the exceptional simple Lie algebras, specific of this characteristic, are obtained in this way from J-ternary algebras coming from structurable algebras and, in particular, a new magic square of Lie superalgebras is constructed, with entries depending on a pair of composition algebras.