One Parameter Commutative Octonions

Year 2023, Volume: 11 Issue: 2, 169 - 175, 31.10.2023

Göksal Bilgici

Abstract

Hyperbolic numbers had been developed in the 19th century. Octonions forms a noncommutative and nonassociative normed division algebra over reals. Octonions have many applications in fields of physics such as quantum logic and string theory. Cayley-Dickson process is applied to quaternions in order to construct octonions and in a sense, we follow a similar process. The aim of this study is to introduce the concept of commutative octonions. We construct this algebra by using some matrix methods. After construction, we give a number of properties of commutative octonions such as fundamental matrices and principal conjugates. We also obtain representation of a commutative octonion as decomposed form, holomorphic and analytic functions of commutative octonions.

Keywords

Commutative octonions , Segre , holomorphic functions

References

• [1] Bilgici, G., Unal, Z., Tokeser, U. and Mert, T., On Fibonacci and Lucas generalized octonions, Ars Combinatoria, 138 (2018), 35–44.
• [2] Catoni, F., Cannata, R., Catoni, V. and Zampetti, P., N-dimensional geometries generated by hypercomplex numbers, Adv. Appl. Clifford Algebr., 15 (2005), 1–25.
• [3] Catoni, F., Cannata, R. and Zampetti, P., An introduction to commutative quaternions, Adv. Appl. Clifford Algebr., 16 (2006), 1–28.
• [4] Catoni, F., Boccalett, D., Cannata, R., Catoni, V., Nichelatti, E. and Zampetti, P., The Mathematics of Mikowski Space–Time and Introduction to Commutative Hypercomplex Numbers. Birkhauser–Verlag, Basel, 2008.
• [5] Freedman, M., Shokrian-Zini, M. and Wang, Z., Quantum computing with octonions, Peking Math. J., 2(3) (2019), 239–273.
• [6] Gunaydin, M., Kallosh, R., Linde, A. and Yamada, Y., M-theory cosmology, octonions, error correcting codes, J. High Energ. Phys., 2021(1) (2021) , 1–60.
• [7] Hamilton, W.R., Lectures on Quaternions, Hodges and Smith, Dublin, 1853.
• [8] Klco, P., Kollarik, M. and Tatar, M., Novel computer algorithm for cough monitoring based on octonions, Respiratory Physiology & Neurobiology, 257 (2018), 36–41.
• [9] Kosal, H.H. and Tosun, M., Commutative quaternion matrices, Adv. Appl. Clifford Algebr., 24 (2014), 769–779.
• [10] Okubo, S., Introduction to Octonion and Other Non-Associative Algebras in Physics, Cambridge University Press, London, 1995.
• [11] Segre, C., The real representations of complex elements and extension to bicomplex system, Math. Ann., 40 (1892), 413–467.
• [12] Singh, T.P., Octonions, trace dynamics and noncommutative geometry-A case for unification in spontaneous quantum gravity, Zeitschrift f¨ur Naturforschung A, 75(12) (2020), 1051–1062.
• [13] Srivastava, G., Gupta, R. Kumar, R. and Le, D.N., Space-time code design using quaternions, octonions and other non-associative structures, International Journal of Electrical and Computer Engineering Systems 10(2) (2019), 91–95.
• [14] Tokeser, U., Mert, T., Unal, Z. and Bilgici, G., On Pell and Pell–Lucas generalized octonions, Turkish Journal of Mathematics and Computer Sciences, 13(2) (2021), 226–233.
• [15] Tokeser, U., Mert, T. and Dundar, Y., Some properties and Vajda theorems of split dual Fibonacci and split dual Lucas octonions, AIMS Math., 7(5) (2022), 8645–8653.
• [16] Weng, Z.H., Frequencies of astrophysical jets and gravitational strengths in the octonion spaces, International Journal of Modern Physics D., 31 (4) (2022), 2250024-1–2250024-16.