Characteristics of Regular Functions Defined on a Semicommutative Subalgebra of 4-Dimensional Complex Matrix Algebra

Abstract

In this paper, we give an extended quaternion as a matrix form involving complex components. We introduce a semicommutative subalgebra $ℂ\left({ℂ}^2\right)$ of the complex matrix algebra $M\left(4,ℂ\right)$ . We exhibit regular functions defined on a domain in ${ℂ}^4$ but taking values in $ℂ\left({ℂ}^2\right)$ . By using the characteristics of these regular functions, we propose the corresponding Cauchy–Riemann equations. In addition, we demonstrate several properties of these regular functions using these novel Cauchy–Riemann equations. Mathematical Subject Classification is 32G35, 32W50, 32A99, and 11E88.