eesti teaduste
akadeemia kirjastus
SINCE 1952
Proceeding cover
of the estonian academy of sciences
ISSN 1736-7530 (Electronic)
ISSN 1736-6046 (Print)
Impact Factor (2021): 1.024
Algebra with ternary cyclic relations, representations and quark model; pp. 61–76

Viktor Abramov, Stefan Groote, Priit Lätt

We propose a unital associative algebra, motivated by a generalization of the Pauli’s exclusion principle proposed for the quark model. The generators of this algebra satisfy the following relations: The sum of squares of all generators is equal to zero (binary relation) and the sum of cyclic permutations of the factors in any triple product of generators is equal to zero (ternary relations). We study the structure of this algebra and calculate the dimensions of spaces spanned by homogeneous monomials. We establish a relation between our algebra and the irreducible representations of the rotation group. In particular, we show that the 10-dimensional space spanned by triple monomials is the space of a double irreducible unitary representation of the rotation group. We use ternary q- and -commutators, where q, are primitive 3rd order roots of unity, to split the 10-dimensional space spanned by triple monomials into a direct sum of two 5-dimensional subspaces. We endow these subspaces with a Hermitian scalar product by means of an orthonormal basis of triple monomials. In each subspace, there is an irreducible unitary representation so(3) → su(5). We calculate the matrix of this representation. The structure of this matrix indicates a possible connection between our algebra and the Georgi–Glashow model.


1. Abramov, V., Kerner, R. and Le Roy, B. Hypersymmetry: A Z3-generalization of supersymmetry. J. Math. Phys., 1997, 38(3), 1650–1669.

2. Abramov, V., Kerner, R. and Liivapuu, O. Algebras with ternary composition law combining Z2 and Z3 gradings. In Springer Proceedings in Mathematics and Statistics (Silvestrov, S., Malyarenko, A. and Rancic, M., eds). Springer, Cham, 2020, 13–45.

3.Abramov, V. Ternary algebras associated with irreducible tensor representations of SO(3) and quark model. 2020. arXiv:2006.05237 

4. Bagger, J. and Lambert, N. Modeling multiple M2’s. Phys. Rev., 2007, D75, 045020. arXiv:hep-th/0611108

5. Bagger, J. and Lambert, N. Gauge symmetry and supersymmetry of multiple M2-branes. Phys. Rev., 2008, D77, 065008. arXiv:0711.0955 [hep-th]

6. Cantarini, N. and Kac, V. Classification of simple linearly compact n-Lie superalgebras. Commun. Math. Phys., 2010, 298, 833–853.

7. Cherkis, S. and Sämann, C. Multiple M2-branes and generalized 3-Lie algebras. Phys. Rev. D, 2008, 78, 066019.

8. Croon, D., Gonzalo, T. E., Graf, L., Košnik, N. and White, G. GUT Physics in the Era of the LHC. Front. Phys., 2019.

9. Dzhumadildaev, A. S. Identities and derivations for Jacobian algebras, Contemp. Math., 2002, 315, 245–278.

10. Filippov, V. T. n-Lie algebras. Siberian Math. J., 1985, 26, 879–891.

11. Groote, S. and Saar, R. Group theory aspects of chaotic strings. In Proceedings of the Conference "QQQ12 – 3Quantum: Algebra, Geometry and Information" (Paal, E., ed.), Tallinn, Estonia, 10–13 July 2012. Institute of Physics Publishing, 2014.

12. Gelfand, I. M., Minlos, R. A. and Shapiro, Z. Ya. Representations of the Rotation and Lorentz Groups and Their Applications. Dover Publications, Mineola, New York, 2018.

13. Kerner, R. Graduation Z3 et la racine cubique de l’opérateur de Dirac. C. R. Acad. Sci. Paris, 1991, 312, 191–196 (in French). 

14. Kerner, R. Z3 graded algebras and the cubic root of the supersymmetry translations. J. Math. Phys., 1992, 33, 403–411.

15. Kerner, R. Ternary generalization of Pauli’s principle and the Z6-graded algebras. Phys. At. Nucl., 2017, 80, 522–534.

16. Kerner, R. The quantum nature of Lorentz invariance. Universe, 2019, 5(1).

17. Nambu, Y. Generalized Hamiltonian mechanics. Phys. Rev. D, 1973, 7, 2405–2412.

Back to Issue