Classification of Low Dimensional 3-Lie Superalgebras

TL;DR

This paper classifies low-dimensional 3-Lie superalgebras, introduces a method to construct higher-order superalgebras via supertrace, and explicitly describes 3-Lie superalgebras derived from Clifford algebras.

Contribution

It provides a classification of low-dimensional 3-Lie superalgebras and introduces a new construction method using supertrace and Clifford algebras.

Findings

Classification of low-dimensional 3-Lie superalgebras.

Construction of (n+1)-Lie superalgebras from n-Lie superalgebras with supertrace.

Explicit description of 3-Lie superalgebras via Clifford algebras.

Abstract

A notion of n-Lie algebra introduced by V.T. Filippov can be viewed as a generalization of a concept of binary Lie algebra to the algebras with n-ary multiplication law. A notion of Lie algebra can be extended to Z_2-graded structures giving a notion of Lie superalgebra. Analogously a notion of n-Lie algebra can be extended to Z_2-graded structures by means of a graded Filippov identity giving a notion of n-Lie superalgebra. We propose a classification of low dimensional 3-Lie superalgebras. We show that given an n-Lie superalgebra equipped with a supertrace one can construct the (n+1)-Lie superalgebra which is referred to as the induced (n+1)-Lie superalgebra. A Clifford algebra endowed with a Z_2-graded structure and a graded commutator can be viewed as the Lie superalgebra. It is well known that this Lie superalgebra has a matrix representation which allows to introduce a supertrace. We apply the method of induced Lie superalgebras to a Clifford algebra to construct the 3-Lie superalgebras and give their explicit description by ternary commutators.