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A Z3 symmetric generalization of the Dirac equation was proposed in recentseries of papers, where its properties and solutions discussed. The generalizedDirac operator acts on "coloured spinors" composed out of six Pauli spinors,describing three colours and particle-antiparticle degrees of freedomcharacterizing a single quark state, thus combining Z2 x Z_2 x Z_3 symmetriesof 12-component generalized wave functions. Spinorial representation of theZ3-graded generalized Lorentz algebra was introduced, leading to the appearanceof extra Z2 x Z2 x Z3 symmetries, probably englobing the symmetries of isospin,flavors and families. The present article proposes a construction of Z3-gradedextension of the Poincar\'e algebra. It turns out that such a generalizationrequires introduction of extended 12-dimensional Minkowskian space-timecontaining the usual 4-dimensional space-time as a subspace, and two othermutually conjugate "replicas" with complex-valued vectors and metric tensors.Representation in terms of differential operators and generalized Casimiroperators are introduced and their symmetry properties are briefly discussed.

The Z3-graded extension of the Poincar\'e algebra