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 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.Recently, there was markedly increased interest of theoretical physics towards the algebras with n-ary multiplication law. Due to the fact that the Lie algebras play a crucial role in theoretical physics, it seems that development of n-ary analog of a concept of Lie algebra is especially important. In [5] V.T. Filippov proposed a notion

Structure and cohomology of 3-Lie-Rinehart superalgebras

“…The structure of 3-Lie algebras induced by Lie algebras, classification of 3-Lie algebras and application to constructions of B.R.S. algebras have been considered in [1,3,4]. Interesting constructions of ternary Lie superalgebras in connection to superspace extension of Nambu-Hamilton equation is considered in [5].…”

Generalized Derivations and Rota-Baxter Operators of $$\varvec{n}$$-ary Hom-Nambu Superalgebras

“…The structure of 3-Lie algebras induced by Lie algebras, classification of 3-Lie algebras and application to constructions of B.R.S. algebras have been considered in [4,5,7].…”

“…This completes the proof.Example 4.10. Consider a 3-dimensional 3-Lie superalgebra (A = A 0 ⊕ A 1 , [•, •, •]) (see[5]), where A 0 is generated by < e 1 > and A 1 is generated by < e 2 , e 3 > and the only non-trivial bracket is[e 2 , e 2 , e 2 ] = e 3 . Define an even linear map α : A → A by α(e 1 ) = ae 1 , α(e 2 ) = be 2 α(e 3 ) = b 3 e 3 , where a, b ∈ K .…”

Structure and cohomology of 3-Lie-Rinehart superalgebras

“…The structure of 3-Lie algebras induced by Lie algebras, classification of 3-Lie algebras and application to constructions of B.R.S. algebras have been considered in [1,3,4]. Interesting constructions of ternary Lie superalgebras in connection to superspace extension of Nambu-Hamilton equation is considered in [5].…”