Degenerate Odd Poisson Bracket on Grassmann Variables

Abstract

A linear degenerate odd Poisson bracket (antibracket) realized solely on Grassmann variables is presented. It is revealed that this bracket has at once three nilpotent $Δ$-like differential operators of the first, the second and the third orders with respect to the Grassmann derivatives. It is shown that these $Δ$-like operators together with the Grassmann-odd nilpotent Casimir function of this bracket form a finite-dimensional Lie superalgebra.