$Z_3$-graded analogues of Clifford algebras and generalization of supersymmetry

TL;DR

This paper introduces ternary analogues of Clifford algebras, explores their algebraic properties, and applies them to develop a $Z_3$-graded extension of supersymmetry, broadening the mathematical framework of supersymmetric theories.

Contribution

It defines ternary Clifford algebras, proves their structural properties, and constructs a $Z_3$-graded supersymmetry generalization using these algebras.

Findings

Ternary Clifford algebra with N generators is isomorphic to a subalgebra of the N+1 generator case.

Derived the ternary commutator of cubic matrices from the algebra.

Constructed a $Z_3$-graded supersymmetry algebra using the ternary Clifford algebra.

Abstract

We define and study the ternary analogues of Clifford algebras. It is proved that the ternary Clifford algebra with $N$ generators is isomorphic to the subalgebra of the elements of grade zero of the ternary Clifford algebra with $N+1$ generators. In the case $N=3$ the ternary commutator of cubic matrices induced by the ternary commutator of the elements of grade zero is derived. We apply the ternary Clifford algebra with one generator to construct the $Z_3$-graded generalization of the simplest algebra of supersymmetries.