INTERNAL SYMMETRY GROUPS OF CUBIC ALGEBRAS

Abstract

We investigate certain Z3-graded associative algebras with cubic Z3 invariant constitutive relations, introduced by one of us some time ago. The invariant forms on finite algebras of this type are given in the cases with two and three generators. We show how the Lorentz symmetry represented by the SL (2, C) group can be introduced without any notion of metric, just as the symmetry of Z3-graded cubic algebra and its constitutive relations. Its representation is found in terms of the Pauli matrices. The relationship of such algebraic constructions with quark states is also considered.