Linear Odd Poisson Bracket on Grassmann Algebra
A linear odd Poisson bracket realized solely in terms of Grassmann variables is suggested. It is revealed that with the bracket, corresponding to a semi-simple Lie group, both a Grassmann-odd Casimir function and invariant (with respect to this group) nilpotent differential operators of the first, second and third orders are naturally related and enter into a finite-dimensional Lie superalgebra. A connection of the quantities, forming this Lie superalgebra, with the BRST charge, $Δ$-operator and ghost number operator is indicated.
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