Linear Odd Poisson Bracket on Grassmann Variables

TL;DR

This paper introduces a linear odd Poisson bracket using Grassmann variables, revealing a new structure with three nilpotent differential operators forming a Lie superalgebra, extending the understanding of antibrackets in mathematical physics.

Contribution

It presents a novel realization of the odd Poisson bracket with three nilpotent operators, contrasting with the canonical second-order operator, and uncovers a related Lie superalgebra structure.

Findings

The bracket corresponds to a semi-simple Lie group.

It has three Grassmann-odd nilpotent differential operators of different orders.

These operators, along with a Casimir function, form a finite-dimensional Lie superalgebra.

Abstract

A linear odd Poisson bracket (antibracket) realized solely in terms of Grassmann variables is suggested. It is revealed that the bracket, which corresponds to a semi-simple Lie group, has at once three Grassmann-odd nilpotent $\Delta$-like differential operators of the first, the second and the third orders with respect to Grassmann derivatives, in contrast with the canonical odd Poisson bracket having the only Grassmann-odd nilpotent differential $\Delta$-operator of the second order. It is shown that these $\Delta$-like operators together with a Grassmann-odd nilpotent Casimir function of this bracket form a finite-dimensional Lie superalgebra.