Towards a $Z_3$-graded approach to quarks' symmetries

Colour $SU(3)$ group is an exact symmetry of Quantum Chromodynamics, which describes strong interactions between quarks and gluons. Supplemented by two internal symmetries, $SU(2)$ and $U(1)$, it serves as the internal symmetry of the Standard Model, describing as well the electroweak interactions of quarks and leptons. The colour$SU(3)$ symmetry is exact, while two other symmetries are broken by means of the Higgs-Kibble mechanism. The three colours and fractional quarks charges with values $1\mathrm{/}3$ and $2\mathrm{/}3$ suggest that the cyclic group $Z_3$ may play a crucial role in quark field dynamics. In this paper we consequently apply the $Z_3$ symmetry to field multiplets describing colour quark fields. Generalized Dirac equation for coloured $12$-component spinors is introduced and its properties are discussed. Imposing $Z_3$-graded Lorentz and Poincar'e covariance leads to enlargement of quark fields multiplets and incorporates additional $Z_2\times Z_3$ symmetry which leads to the appearance of three generations (families) of distinct quark doublets.