Ternary algebras with $Z_2$ and $Z_3$ grading and generalized representations of the Lorentz group

We discuss cubic and ternary algebras which are a direct generalization of Grassmann  and Clifford algebras, but with $Z_3$-grading replacing the usual $Z_2$-grading. Combining $Z_2$ and $Z_3$ gradings results in algebras with $Z_6$ grading, which are also investigated.

Elementary properties and structures of such algebras are discussed, with special interest in low-dimensional ones, with two or three generators.

Invariant antisymmetric quadratic and cubic forms on such algebras are introduced, and it is shown how the  $SL(2,C)$} group arises naturally in the case of lowest dimension, with two generators only, as the symmetry group preserving these forms.

In the case of lowest dimension, with two generators only, it is shown how the cubic combinations of $Z_3$-graded elements behave like Lorentz spinors, and the binary product of elements of this algebra with an element of the conjugate algebra behave like Lorentz vectors.

Then Pauli's principle is generalized for the case of the $Z_3$ graded ternary algebras leading to cubic commutation relations. A generalized Dirac equation is introduced.

The model suggests the origin of the color $SU(3)$ symmetry of strong interactions.