Matrix 3-Lie superalgebras and BRST supersymmetry

Abstract

Given a matrix Lie algebra one can construct the 3-Lie algebra by means of the trace of a matrix. In the present paper we show that this approach can be extended to the infinitedimensional Lie algebra of vector fields on a manifold if instead of the trace of a matrix we consider a differential 1-form which satisfies certain conditions. Then we show that the same approach can be extended to matrix Lie superalgebras gl (m, n) if instead of the trace of a matrix we make use of the super trace of a matrix. It is proved that a graded triple commutator of matrices constructed with the help of the graded commutator and the super trace satisfies a graded ternary Filippov-Jacobi identity. In two particular cases of gl (1, 2) and gl(2, 2) we show that the Pauli and Dirac matrices generate the matrix 3-Lie superalgebras, and we find the non-trivial graded triple commutators of these algebras. We propose a Clifford algebra approach to 3-Lie superalgebras induced by Lie superalgebras. We also discuss an application of matrix 3-Lie superalgebras in BRST-formalism.

Quantum Super Nambu Bracket of Cubic Supermatrices and 3-Lie Superalgebra

“…(A * j B) * j C = A * j (B * j C). Thus the multiplication of 3-dimensional matrices (5.2) induces the structure of unital associative algebra on the vector space of nth order cubic matrices M (3) n , where the identity element of this algebra is the nth order cubic matrix I, whose any section of orientation (j) is the nth order square unit matrix E, i.e. I (j) r = E and r = 1, 2, .…”

“…n equipped with the binary multiplication of cubic matrices is the unital associative algebra and consequently the commutator of two cubic matrices A, B ∈ M n endowed with the commutator becomes the Lie algebra, which we will denote by gl (3) n . It is easy to show that the trace of a cubic matrix introduced in the previous section (4.2) vanishes on the commutator of two cubic matrices.…”

“…In the paper [5] the authors propose the realization of quantum Nambu bracket given by the formula where X, Y, Z are either square matrices or cubic matrices, and prove that this quantum Nambu bracket satisfies the Filippov-Jacobi identity. Inspired by (1.1) in [3] we extend this approach to Z 2 -graded case by assuming that X, Y, Z are supermatrices and taking supertrace of a matrix instead of the trace and Z 2 -graded commutator instead of ordinary commutator, i.e. we define the graded triple commutator of supermatrices by…”

Nambu–Poisson bracket on superspace

“…In the paper [3] the authors propose the realization of quantum Nambu-Poisson bracket by means of nth order matrices, where the triple commutator is defined with the help of usual commutator and the trace of a matrix. This approach is extended to super Nambu-Poisson bracket by means of supermatrices, where the Z 2 -graded triple commutator is defined with the help of the supertrace of a supermatrix [1], [2]. A smooth manifold M endowed with a Poisson bracket is referred to as a Poisson manifold.…”

“…We can prove a similar theorem for the whole n-ary Ψ-bracket (13). (1) The property of totally skew-symmetry, which means that any permutation of arguments of n-ary Ψ-bracket changes its sign according to the parity of permutation; (2) The Leibniz rule for a product of functions, i.e.…”