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Hypersymmetry: A $$\mathbb{Z}_ 3$$-graded generalization of supersymmetry. (English) Zbl 0872.58006
Summary: We propose a generalization of non-commutative geometry and gauge theories based on ternary $$\mathbb{Z}_3$$-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 $$\mathbb{Z}_3$$-group, which may be either trivial, i.e., $$abc=bca=cab$$, generalizing the usual commutativity, or non-trivial, i.e., $$abc =jbca$$, with $$j=e^{(2\pi i)/3}$$. The usual $$\mathbb{Z}_2$$-graded structures such as Grassmann, Lie, and Clifford algebras are generalized to the $$\mathbb{Z}_3$$-graded case. Certain suggestions concerning the eventual use of these new structures in physics of elementary particles and fields are exposed.

##### MSC:
 46L85 Noncommutative topology 46L87 Noncommutative differential geometry 81Q60 Supersymmetry and quantum mechanics
##### Keywords:
hypersymmetry; $$\mathbb{Z}_ 3$$-graded structures
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##### References:
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