Hypersymmetry: A ℤ3-graded generalization of supersymmetry

Source abstract

We propose a generalization of non-commutative geometry and gauge theories based on ternary ℤ3-graded structures. In the new algebraic structures we define, all products of two entities are left free, the only constraining relations being imposed on ternary products. These relations reflect the action of the ℤ3-group, which may be either trivial, i.e., abc=bca=cab, generalizing the usual commutativity, or non-trivial, i.e., abc=jbca, with j=e(2πi)/3. The usual ℤ2-graded structures such as Grassmann, Lie, and Clifford algebras are generalized to the ℤ3-graded case. Certain suggestions concerning the eventual use of these new structures in physics of elementary particles and fields are exposed.