Regular Functions with Values in a Noncommutative Algebra using Clifford Analysis

Su Jin Lim, Kwang Ho Shon

We construct a noncommutative algebra $C(2)$ that is a subalgebra of the Pauli matrices of $M(2;C)$, and investigate the properties of solutions with values in $C(2)$ of the inhomogeneous Cauchy--Riemann system of partial differential equations with coefficients in the associated Pauli matrices. In addition, we construct a commutative subalgebra $C(4)$ of $M(4;C)$, obtain some properties of biregular functions with values in $C(2)$ on $\Omega$ in $C^2\times C^2$, define a J-regular function of four complex variables with values in C(4), and examine some properties of J-regular functions of partial differential equations.