Three-algebras, triple systems and 3-graded Lie superalgebras

TL;DR

This paper establishes a correspondence between three-algebras used in certain supersymmetric theories and Lie superalgebras, providing a classification framework based on known Lie superalgebra classifications.

Contribution

It demonstrates that three-algebras can be described as generalized Jordan triple systems, linking them to Lie superalgebras and enabling classification of simple three-algebras.

Findings

Three-algebras correspond to Lie superalgebras

Simple three-algebras are classified by simple Lie superalgebras

Provides a natural framework for understanding three-algebras in supersymmetric theories

Abstract

The three-algebras used by Bagger and Lambert in N=6 theories of ABJM type are in one-to-one correspondence with a certain type of Lie superalgebras. We show that the description of three-algebras as generalized Jordan triple systems naturally leads to this correspondence. Furthermore, we show that simple three-algebras correspond to simple Lie superalgebras, and vice versa. This gives a classification of simple three-algebras from the well known classification of simple Lie superalgebras.