Generalized Quaternions and Matrix Algebra

Yıl 2023, Cilt: 23 Sayı: 3, 638 - 647, 28.06.2023

Erhan Ata , Ümit Ziya Savcı

https://doi.org/10.35414/akufemubid.1182145

Öz

In this paper, we established the connection between generalized quaternion algebra and real (complex) matrix algebras by using Hamilton operators. We obtained real and complex matrices corresponding to the real and complex basis of the generalized quaternions. Also, we investigated the basis features of real and complex matrices. We get Pauli matrices corresponding to generalized quaternions. Then, we have shown that the algebra produced by these matrices is isomorphic to the Clifford algebra Cl(E_αβ^3) produced by generalized space E_αβ^3.
Finally, we studied the relations among the symplectic matrices group corresponding to generalized unit quaternions, generalized unitary matrices group, and generalized orthogonal matrices group.

Anahtar Kelimeler

Generalized quaternions;, ; Hamilton operators, Pauli matices, Generalized unitary matrices, Generalized orthogonal matrices

Genelleştirilmiş Kuaterniyonlar ve Matris Cebiri

Öz

Bu çalışmada, Hamilton operatörlerini kullanarak genelleştirilmiş kuaterniyon cebiri ile gerçek (kompleks) matris cebirleri arasındaki bağlantıyı kurduk. Genelleştirilmiş kuaterniyonların gerçel ve kompleks temeline karşılık gelen gerçel ve kompleks matrisler elde ettik. Ayrıca, gerçek ve kompleks matrislerin temel özelliklerini araştırdık. Genelleştirilmiş kuaterniyonlara karşılık gelen Pauli matrislerini elde ettik. Daha sonra, bu matrisler tarafından üretilen cebirin, genelleştirilmiş E_αβ^3 uzayı tarafından üretilen Clifford cebiri Cl(E_αβ^3) ile izomorf olduğunu gösterdik.
Son olarak, genelleştirilmiş birim kuaterniyonlara karşılık gelen simplektik matrisler grubu, genelleştirilmiş birim matrisler grubu ve genelleştirilmiş ortogonal matrisler grubu arasındaki ilişkileri inceledik.

Anahtar Kelimeler

Genelleştirilmiş kuaterniyonlar, Hamilton operatörleri, Pauli matrisleri, Genelleştirilmiş üniter matrisler, Genelleştirilmiş ortogonal matrisler

Kaynakça

• Alagoz, Y. Oral, K.H. and Yuce, S., 2012. Split quaternion matrices. Miskolc Mathematical Notes, 13(2), 223–232.
• Aragon, G., Aragon J.L. and Rodriguez, M.A., 1997. Clifford algebras and geometric Algebra. Adv. Appl. Clifford Al., 7(2), 91–102.
• Ata, E. and Yaylı, Y., 2009. Split quaternions and semi-Euclidean projective spaces. Chaos, Solitons and Fractals, 41(4), 1910–1915.
• Ata ,E., Kemer, Y. and Atasoy, A., 2012. Quadratic Formulas for Generalized Quaternions. Dumlupınar ¨Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 28, 27–34.
• Ata, E. and Yıldırım, Y., 2018. A Different Polar Representation for Generalized and Generalized Dual Quaternions. Adv. Appl. Clifford Al., 28(4), 77.
• [Ata, E., Savcı, Ü.Z., 2021. Spherical kinematics in 3-dimensional generalized space. International Journal of Geometric Methods in Modern Physics, 18(3), 2150033.
• Catoni, F., Cannata, R., Catoni, V. and Zampetti, P., 2005. N-dimensional geometries generated by hypercomplex numbers. Advances in Applied Clifford Algebras, 15(1), 1–25.
• Cockle, J., 1849. On Systems of Algebra Involving More than One Imaginary. Philos. Mag. (series 3), 35, 434–435.
• Grob, J., Trenkler, G. and Troschke, S.O., 2003. Quaternions: further contributions to a matrix oriented approach. Linear Algebra Appl., 326(2), 251–255.
• Hamilton, W.R., 1853. Lectures on quaternions, Landmark Writings in Western Mathematics.
• Hamilton, W.R., 1866. Elements of quaternions, Longmans, Green and Company.
• Jafari, M., 2012 Generalized hamilton operators and lie groups, Ph.D. Thesis, Ankara University, .
• Jafarı, M. and Yaylı, Y., 2015. Generalızed Quaternions and Rotation in 3-Space E3αβ. TWMS J. Pure Appl. Math., 6(2), 224–232.
• Lam, T.Y., 2005. Introduction to Quadratic Forms Over Fields, American Mathematical Society, USA.
• Özdemir, M. A. and Ergin, A., 2006. Rotations with unit timelike quaternions in Minkowski 3-space. Journal of geometry and physics, 56(2), 322–336.
• Sangwine, S.J. and Le Bihan, N., 2010. Quaternion polar representation with a complex modulus and complex argument inspired by the Cayley-Dickson form. Adv. Appl. Clifford Al., 20(1), 111–120.