Appears in collection : Combinatorics and Arithmetic for Physics - 2020

A generalization of Dirac’s equation is presented, incorporating the three-valued colour variable in a way which makes it intertwine with the Lorentz transformations. We show how the Lorentz-Poincaré group must be extended to accomodate both $SU(3)$ and the Lorentz transformations. Both symmetries become intertwined, so that the system can be diagonalized only after the sixth iteration, leading to a six-order characteristic equation with complex masses similar to those of the Lee-Wick model. The spinorial representation of the $\mathbb Z_3$-graded Lorentz algebra is presented, and its vectorial counterpart acting on a $\mathbb Z_3$-graded extension of the Minkowski space-time is also constucted. Application to new formulation of the QCD and its gauge-field content is briefly evoked.