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 infinite-dimensional 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 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 and 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.