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

International audienceColour 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 1/3 and 2/3 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 Poincaré covariance leads to enlargement of quark fields multiplets and incorporates additional Z 2 × Z 3 symmetry which leads to the appearance of three generations (families) of distinct quark doublets