  We investigate certain $Z_3$-graded associative algebras with cubic
$Z_3$-invariant constitutive relations. The invariant forms on finite algebras
of this type are given in the low dimensional cases with two or three
generators.
  We show how the Lorentz symmetry represented by the $SL(2, {\bf C})$ group
emerges naturally without any notion of Minkowskian metric, just as the
invariance group of the $Z_3$-graded cubic algebra and its constitutive
relations. Its representation is found in terms of Pauli matrices. The
relationship of this construction with the operators defining quark states is
also considered, and a third-order analogue of the Klein-Gordon equation is
introduced. Cubic products of its solutions may provide the basis for the
familiar wave functions satisfying Dirac and Klein-Gordon equations.
