Generalized Quaternions and Matrix Algebra

Year 2023, Volume: 23 Issue: 3, 638 - 647, 28.06.2023

Erhan Ata , Ümit Ziya Savcı

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

Abstract

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.

Keywords

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

Genelleştirilmiş Kuaterniyonlar ve Matris Cebiri

Abstract

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.

Keywords

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

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