Hypersymmetry: a Z_3-graded generalization of supersymmetry

TL;DR

This paper introduces a Z_3-graded algebraic framework generalizing supersymmetry and non-commutative geometry, with potential applications in elementary particle physics.

Contribution

It develops a novel Z_3-graded algebraic structure that extends traditional Z_2-graded systems like Grassmann and Clifford algebras, proposing new mathematical tools for physics.

Findings

Defined ternary Z_3-graded algebraic relations

Generalized Grassmann, Lie, and Clifford algebras to Z_3-graded case

Suggested potential applications in elementary particle physics

Abstract

We propose a generalization of non-commutative geometry and gauge theories based on ternary Z_3-graded structures. In the new algebraic structures we define, we leave all products of two entities free, imposing relations on ternary products only. These relations reflect the action of the Z_3-group, which may be either trivial, i.e. abc=bca=cab, generalizing the usual commutativity, or non-trivial, i.e. abc=jbca, with j=e^{(2\pi i)/3}. The usual Z_2-graded structures such as Grassmann, Lie and Clifford algebras are generalized to the Z_3-graded case. Certain suggestions concerning the eventual use of these new structures in physics of elementary particles are exposed.