The cubic chessboard

We present a survey of recent results, scattered in a series of papers that have appeared during the past five years, whose common denominator has been the use of cubic relations in various algebraic structures.

Cubic (or ternary) relations can represent different symmetries with respect to the permutation group S₃, or its cyclic subgroup Z₃. Also ordinary or ternary algebras can be divided into different classes with respect to their symmetry properties. We pay special attention to the non-associative ternary algebra of 3-forms (or cubic matrices), and Z₃-graded matrix algebras.

We also discuss the Z₃-graded generalization of Grassmann algebras and their realization in generalized exterior differential forms dξ and d²ξ, with d³ξ=0. A new type of gauge theory based on this differential calculus is presented.

Finally, a ternary generalization of Clifford algebras is introduced, and an analogue of Dirac's equation is discussed, which can be diagonalized only after taking the cube of the Z₃-graded generalization of Dirac's operator. A possibility of using these ideas for the description of quark fields is suggested and discussed in the last section.