Algebras with Ternary Composition Law Combining $\mathrm {Z_2}$ and $\mathrm {Z_3}$ Gradings

Abstract

We investigate the possibility of combining the usual Grassmann algebras with their ternary $\mathbb Z_3$-graded counterparts, thus creating a more general algebra with quadratic and cubic constitutive relations coexisting together. 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 those cases. Then we study a particular case of algebras generated by two types of variables, $\xi ^{\alpha }$ and $\theta ^A$, satisfying quadratic and cubic relations respectively, $\xi ^{\alpha } \xi ^{\beta } = - \xi ^{\beta } \xi ^{\alpha }$ and $\theta ^A \theta ^B \theta ^C = j \theta ^B \theta ^C \theta ^A$, $j = e^{\frac{2 \pi i}{3}}$. Differential calculus of the first order is defined on these algebras, and its fundamental properties investigated. The invariance properties of the generalized algebras are also considered.