Hypersymmetry: A ℤ3-graded generalization of supersymmetry

Abstract

We propose a generalization of non-commutative geometry and gauge theories based on ternary Z 3 -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 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 (2πi)/3 . The usual Z 2 -graded structures such as Grassmann, Lie and Clifford algebras are generalized to the Z 3 -graded case. Certain suggestions concerning the eventual use of these new structures in physics of elementary particles are exposed.

Ulam Stability Generalizations of 4th- Order Ternary Derivations Associated to a Jmrassias Quartic Functional Equation on Fréchet Algebras

“…The ternary groupoid (G, [., ., . ]) is called commutative if [x 1 , x 2 , x 3 ] = [x δ (1) , x δ (2) , x δ (3) ] for all x 1 , x 2 , x 3 ∈ G and all permutations δ of {1, 2, 3}. If a binary operation • is defined on G such that [x, y, z] = (x • y) • z for all x, y, z ∈ G, then we say that [., ., .]…”

Generalized ulam-hyers stability of C*-Ternary algebra n-Homomorphisms for a functional equation

“…(1) The algebra of nonions generated by two matrices ⎛ was introduced by Sylvester as a ternary analog of Hamilton's quaternions [4]. (2) The quark model inspired a particular brand of ternary algebraic systems.…”

“…There are also some applications, although still hypothetical, in the fractional quantum Hall effect, the non-standard statistics, supersymmetric theory, and Yang-Baxter equation [4,6].…”

APPROXIMATE BI-HOMOMORPHISMS AND BI-DERIVATIONS IN C*-TERNARY ALGEBRAS

“…(1) The algebra of 'nonions' generated by two matrices was introduced by Sylvester as a ternary analog of Hamilton's quaternions [1]. (2) The quark model inspired a particular brand of ternary algebraic systems.…”

“…There are also some applications, although still hypothetical, in the fractional quantum Hall effect, the non-standard statistics, supersymmetric theory, and Yang-Baxter equation [1,17,39].…”