Z_n^3-graded colored supersymmetry

TL;DR

This paper introduces Z_{n}^{3}-graded Grassmann algebras with unique properties, proposing their potential use in modeling quark fields through generalized supersymmetry generators that are cubic roots of traditional ones.

Contribution

It constructs new Z_{n}^{3}-graded Grassmann algebras and develops generalized supersymmetry generators based on these algebras, extending the mathematical framework of supersymmetry.

Findings

Algebras are graded by Z_{n}^{3} and contain ordinary Grassmann algebras.

Generalized supersymmetry generators are cubic roots of standard generators.

Potential application to modeling quark fields.

Abstract

We build generalizations of the Grassmann algebras from a few simple assumptions which are that they are graded, maximally symmetric and contain an ordinary Grassmann algebra as a subalgebra. These algebras are graded by Z_{n}^{3} and display surprising properties that indicate their possible application to the modelization of quark fields. We build the generalized supersymmetry generators based on these algebras and their derivation operators. These generators are cubic roots of the usual supersymmetry generators.