Hypersymmetry: a Z3-graded generalization of Supersymmetry, J.Math.Phys

We propose a generalization of non-commutative geometry and gauge theories based on ternary Z3-graded structures. In the new algebraic structures we define, we leave all products of two entities free, imposing relations on ternary products only. 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 are exposed. Proposed PACS classification numbers: 03.65.F, 11.30.P, 12.40.-y, 14.65.-q ∗Permanent Address: Institute of Pure Mathematics, Tartu University, Vanemuise 46, Tartu, Estonia