On one space of four-dimensional numbers

  • A. T. Rakhymova L.N. Gumilyov Eurasian National University, Nur-Sultan, Kazakhstan http://orcid.org/0000-0002-8888-8686
  • M. B. Gabbassov System research company "Factor" , Nur-Sultan, Kazakhstan
  • K. M. Shapen L.N. Gumilyov Eurasian National University, Nur-Sultan, Kazakhstan


The theory of four-dimensional numbers was first introduced by W.R. Hamilton in 1843, which is called the theory of quaternions. In this theory multiplication is a non-commutative operation, as a result of which it was not possible to construct a full-fledged mathematical analysis of functional spaces. In 2003, a new theory of functions of four-dimensional variables was published by Kazakh mathematicians B. Maukeev and M.M. Abenov, where commutative multiplication is introduced, which allows solving three-dimensional models of mechanics by the analytical method. A more complete presentation of the new theory was published in 2019 by M.M. Abenov in a monograph. While developing this theory, Abenov and M.B. Gabbasov found all four-dimensional spaces with commutative multiplication, which were assigned the designations M2 - M7, and it became necessary to study these spaces. This paper studies one of these spaces, namely the space of four-dimensional numbers M5. The purpose of this paper is to study the properties of four-dimensional numbers of the space M5 and substantiate its significance. In this study, new results are obtained on an algebra of the space M5, various norms and metrics are introduced, and the properties of number sequences are considered.


[1] William R. Hamilton, Lectures on Quaternions: containing a systematic statement of a new mathematical method (Dublin University Press, 1853, 868 pp.).

[2] A. Sudbery, "Quaternionic analysis" , Math. Proc. Camb. Phil. Soc. 85(1979), 199-225.

[3] A. Buchman, "A brief history of quaternions and the theory of holomorphic functions of quaternionic variables" , https://www.maa.org/sites/default/files/pdf/upload_library/46/HOMSIGMAA/Buchmann.pdf

[4] Tsit Lam, "Hamilton’s quaternions" , Handbook of Algebra 3(2003), 429-454. DOI: https://doi.org/10.1016/S1570-7954(03)80068-2

[5] D. Eberly, "Quaternion algebra and calculus" , https://www.geometrictools.com/Documentation/Quaternions.pdf

[6] S.L. Alder, "Quaternionic quantum field theory" , Commun. Math. Phys. 104(1986), 611-656.

[7] S.L. Adler, Quaternionic Quantum Mechanics and Quantum Fields (New York: Oxford University Press, 1995).

[8] A. Baker, "Right eigenvalues for quaternionic matrices: A topological approach" , Linear Algebra Appl. 286(1999), 303-309.

[9] A. Brauer, "Limits for the characteristic roots of matrices II" , Duke Math. J. 14(1947), 21-26.

[10] J.L. Brenner, "Matrices of quaternions" , Pac. J. Math. 1(1959), 329-335.

[11] A. Cayley, "On certain results relating to quaternions" , Philos. Mag. 26(1845), 141-145.

[12] L. Chen, "Definition of determinant and Cramer solution over the quaternion field" , Acta Math. Sinica (N.S.) 7(1991), 171-180.

[13] Kantor I.L., Solodovnikov A.S., Giperkompleksnie chisla [Hyper complex numbers] (М.: Nauka, 1973, 144 p.) [in Russian].

[14] A. Skowronski and K. Yamagata, Frobenius Algebras I. Basic Representation Theory (European mathematical society publishing house, 2012, 662 pp.)

[15] Maukeev B.E., Abenov М.М., Nachal’nie glavy teorii funksii bikompleksnogo peremennogo [The initial chapters of the theory of functions of a bicomplex variable] (Almaty: LLP ¾MTIA¿, 2003, 58 pp.) [in Russian].

[16] Abenov M.M., Chetirehmernaya matematika. Metody i prilozheniya. Nauchnaya monographia [Four-dimensional mathematics: Methods and applications. Scientific monograph] (Almaty: Publishing House, 2019, 176 pp.)

[17] L. Hsu, "On symmetric, orthogonal, and skew symmetric matrices" , Proc. Edinburg Math. Soc., ser. 2. 10(1953), 37-44.

[18] R. Bellman, "Notes on matrix theory" , Amer. Math. Monthly 60(1953), 173-175.

[19] Lavrent’ev M.A., Shabat B.V. Metody teorii funksii kompleksnogo peremennogo [Methods of the theory of functions of a complex variable] (М.: Nauka, 1965, 716 pp.) [in Russian].

[20] Bitsadze A.V., Osnovi teorii analyticheskih funksii kompleksnogo peremennogo [Fundamentals of the theory of analytic functions of a complex variable] (М.: Nauka, 1984, 280 pp.) [in Russian].

[21] Kolmogorov A.N., Fomin S.V., Elemenri teorii funksii i funksional’nogo analyza [Elements of function theory and functional analysis] (М.: Nauka, 1989, 624 pp.) [in Russian].

[22] Sidorov V.Yu., Fedoryuk M.I., Shabunin M., Leksii po teorii funksii kompleksnogo peremennogo [Theory of functions of complex variables] (M.: Nauka, 1982, 488 pp.) [in Russian].

[23] Gantmakher F.R., Teoriya matrits [Matrix theory] (М.: Nauka, 1967, 576 pp.) [in Russian].

[24] Gantmakher F.R. and Krein M.G., K stukture ortogonal’noi matrisi [To a structure of orthogonal matrix] (Trudy Fiz.-Mat. Otdela VUAN, Kiev, 1929, P. 1-8) [in Russian].

[25] Abenov M.M., Gabbassov M.B., Anyzotropnie chetirehmernie prostranstva ili novie kvaternioni [Anisotropic four-dimensional spaces or new quaternions] (Preprint, Nur-Sultan, 2020) [in Russian].
How to Cite
RAKHYMOVA, A. T.; GABBASSOV, M. B.; SHAPEN, K. M.. On one space of four-dimensional numbers. Journal of Mathematics, Mechanics and Computer Science, [S.l.], v. 108, n. 4, p. 81-98, dec. 2020. ISSN 2617-4871. Available at: <https://bm.kaznu.kz/index.php/kaznu/article/view/823>. Date accessed: 17 jan. 2021. doi: https://doi.org/10.26577/JMMCS.2020.v108.i4.07.
Keywords four-dimensional number, spectrum, eigenvalue, symplectic module, spectral norm