Quantum super Nambu bracket of cubic supermatrices and 3-Lie superalgebra

Abstract

We construct the graded triple Lie commutator of cubic supermatrices, which we call the quantum super Nambu bracket of cubic supermatrices, and prove that it satisfies the graded Filippov-Jacobi identity of 3-Lie superalgebra. For this purpose we use the basic notions of the calculus of 3-dimensional matrices, define the Z_2-graded (or super) structure of a cubic matrix relative to one of the directions of a cubic matrix and the super trace of a cubic supermatrix. Making use of the super trace of a cubic supermatrix we introduce the triple product of cubic supermatrices and find the identities for this triple product, where one of them can be regarded as the analog of ternary associativity. We also show that given a Lie algebra one can construct the $n$-ary Lie bracket by means of an (n-2)-cochain of given Lie algebra and find the conditions under which this n-ary bracket satisfies the Filippov-Jacobi identity, thereby inducing the structure of n-Lie algebra. We extend this approach to n-Lie superalgebras.