HYPERSYMMETRY - A Z(3)-GRADED GENERALIZATION OF SUPERSYMMETRY

We propose a generalization of non-commutative geometry and gauge theo ries based on ternary Z(3)-graded structures. In the new algebraic str uctures we define, all products of two entities are left free, the onl y constraining relations being imposed on ternary products. These rela tions reflect the action of the Z(3)-group, which may be either trivia l, i.e., abc = bca = cab, generalizing the usual commutativity, or non -trivial, i.e., abe = jbca, with j = e((2 pi i))/3. The usual Z(2)-gra ded structures such as Grassmann, Lie, and Clifford algebras are gener alized to the Z(3)-graded case. Certain suggestions concerning the eve ntual use of these new structures in physics of elementary particles a nd fields are exposed. (C) 1997 American Institute of Physics.