Classification of Low Dimensional 3-Lie Superalgebras

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 $\mathbb Z_2$-graded structures giving a notion of Lie superalgebra. Analogously a notion of n-Lie algebra can be extended to $\mathbb Z_2$-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 $\mathbb Z_2$-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.