×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Kerner R., C.R. Acad. Sci. Paris 312 pp 191– (1991)
[2] DOI: 10.1063/1.529922 · doi:10.1063/1.529922
[3] DOI: 10.1063/1.528917 · Zbl 0704.53082 · doi:10.1063/1.528917
[4] DOI: 10.1088/0264-9381/8/6/008 · Zbl 0719.53063 · doi:10.1088/0264-9381/8/6/008
[5] DOI: 10.1063/1.531526 · Zbl 0864.17002 · doi:10.1063/1.531526
[6] DOI: 10.1063/1.530519 · Zbl 0821.58006 · doi:10.1063/1.530519
[7] DOI: 10.1063/1.531688 · Zbl 0862.15022 · doi:10.1063/1.531688
[8] DOI: 10.1007/BF00714408 · Zbl 0852.58002 · doi:10.1007/BF00714408
[9] DOI: 10.1103/PhysRevD.7.2405 · Zbl 1027.70503 · doi:10.1103/PhysRevD.7.2405
[10] DOI: 10.1007/BF02103278 · Zbl 0808.70015 · doi:10.1007/BF02103278
[11] DOI: 10.1090/S0002-9947-1951-0041118-9 · doi:10.1090/S0002-9947-1951-0041118-9
[12] DOI: 10.1090/S0002-9947-1952-0045702-9 · doi:10.1090/S0002-9947-1952-0045702-9
[13] Sylvester J. J., John Hopkins Univ. Circulars 3 pp 7– (1884)
[14] DOI: 10.1016/0040-9383(86)90007-8 · Zbl 0592.55015 · doi:10.1016/0040-9383(86)90007-8
[15] Abramov V., Groups Geom. 12 pp 201– (1995)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.