Algebras with ternary law of composition combining Z_2 and Z_3 gradings

TL;DR

This paper explores the integration of Grassmann and ternary Z_3-graded algebras, creating a unified algebraic framework with quadratic and cubic relations and analyzing their symmetry properties and invariance groups.

Contribution

It introduces a new algebraic structure combining Z_2 and Z_3 gradings, extending classification and studying invariance groups of these generalized algebras.

Findings

Classification of ternary and cubic algebras under S_3 symmetry

Construction of an algebra with both Grassmann and ternary generators

Identification of the invariance group of the combined algebra

Abstract

In the present article we investigate the possibility of combining the usual Grassmann algebras with their ternary Z_3-graded counterpart, thus creating a more general algebra with coexisting quadratic and cubic constitutive relations. We recall the classification of ternary and cubic algebras according to the symmetry properties of ternary products under the action of the S_3 permutation group. Instead of only two kinds of binary algebras, symmetric or antisymmetric, here we get four different generalizations of each of these two cases. Then we study a particular case of algebras generated by two types of variables, the generators of Grassmann algebra and generators of ternary analog of Grassmann algebra, satisfying quadratic and cubic relations respectively, i.e. the generators of Grassmann algebra anticommute and a triple product of any three generators of a ternary analog of Grassmann algebra is equal to cyclic permutation of generators in this product multiplied by a primitive 3rd root of unity. The invariance group of the generalized algebra is introduced and investigated.