Linear Odd Poisson Bracket on Grassmann Algebra

TL;DR

This paper introduces a linear odd Poisson bracket on Grassmann algebra, revealing its relation to Lie superalgebras, Casimir functions, and operators relevant in BRST symmetry and quantum field theory.

Contribution

It presents a novel realization of an odd Poisson bracket solely in Grassmann variables and explores its connection to Lie superalgebras and BRST-related operators.

Findings

Existence of a Grassmann-odd Casimir function.

Identification of invariant nilpotent differential operators.

Establishment of a finite-dimensional Lie superalgebra structure.

Abstract

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, $\Delta$-operator and ghost number operator is indicated.