Degenerate Odd Poisson Bracket on Grassmann Variables

TL;DR

This paper introduces a specific degenerate odd Poisson bracket on Grassmann variables, revealing its associated nilpotent operators and Lie superalgebra structure, advancing the mathematical understanding of supersymmetric structures.

Contribution

It presents a novel realization of a degenerate odd Poisson bracket on Grassmann variables and uncovers its associated nilpotent operators and Lie superalgebra.

Findings

The bracket has three nilpotent Δ-like differential operators.

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

The structure enhances understanding of supersymmetric algebraic frameworks.

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 $\Delta$-like differential operators of the first, the second and the third orders with respect to the Grassmann derivatives. It is shown that these $\Delta$-like operators together with the Grassmann-odd nilpotent Casimir function of this bracket form a finite-dimensional Lie superalgebra.