Ternary Z₆-graded algebras

Abstract

We investigate the possibility of combining the usual Grassmann algebras with their ternary Z₃-graded counterpart, thus creating a more general algebrawith coexisting quadratic and cubic constitutive relations.We study a particular case of algebras generated by two types of variables, ξ^a and θ^A, satisfying quadratic and cubic relations respectively, ξ^a ξ^b = –ξ^b ξ^a and θ^Aθ^Bθ^C =j θ^Bθ^Cθ^A a$$EQUATION$$ θ A θ B θ C = j² θ ^B θ ^C θ ^A ; with j = e $$EQUATION$$ 2pi 3 . We show how one can combine the Z₂ and the Z₃ gradings of those binary and ternary algebras and merge them into a common Z₆-graded algebra.