#### Abstract

In this paper, we give an extended quaternion as a matrix form involving complex components. We introduce a semicommutative subalgebra of the complex matrix algebra . We exhibit regular functions defined on a domain in but taking values in . 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.

#### 1. Introduction

Introduced by Hamilton in 1894, quaternions form an algebra generated as a noncommutative division (associative) algebra. Quaternions are used in physics and engineering fields such as electromechanics, quantum mechanics, and 3D animation (see [1–3]). To increase the utilization of quaternions and expand their application, several researchers have attempted to modify, supplement, or expand quaternions. In particular, considering that quaternions form a number system extending the complex numbers, there have been several studies already made since the past few decades regarding holomorphic functions of a quaternion variable. These researchers have investigated whether the properties of complex functions and the definition of holomorphic functions are applicable to functions of a quaternion variable. To do this, researchers have defined holomorphic (regular) functions of a quaternion variable and they have investigated their properties. Fueter [4] defined regular functions over the quaternion field identified with . Based on this definition, Fueter investigated a generalization of the Cauchy–Riemann equations in the complex holomorphic function theory. Delanghe [5] established Stokes’ theorem and Cauchy’s and Green’s formulas for functions with values in Clifford algebra over a quadratic -dimensional real vector space with an orthogonal base. Using a generalization of the Cauchy–Riemann equation, Ryan [6] introduced a regularity of quaternion-valued functions and developed a regular function theory of complex Clifford algebra. Sudbery [7] demonstrated basic algebraic properties of quaternions and quaternionic differential forms. Using exterior differential calculus, Sudbery proposed new and simple proofs of most of the main theorems and clarified the relationship between quaternionic analysis and complex analysis. Naser [8] and Nôno [9] introduced certain quaternionic differential operators and defined the hyperholomorphy of quaternionic functions. Nôno and Inenaga [10] developed hyperholomorphic functions of quaternionic variables as holomorphic function theory of . Kim et al. [11, 12] researched the properties of regular functions with values in the three-dimensional real skew field (called ternary field ) and reduced quaternions using Clifford analysis. They proposed the corresponding Cauchy–Riemann equations with applications defined on . Sommen [13] constructed kernels and monogenic and holomorphic functions, leading to connections between the theory of holomorphic functions of several variables and the theory of monogenic functions. Based on these studies, in this paper, we propose a form distinct from the previously attempted expansion of quaternions. In previous studies, extensions of the quaternionic number system and various combinations of quaternions were suggested. Baez [14] applied octonions to spinors, Bott periodicity, projective and Lorentzian geometry, Jordan algebras, and exceptional Lie groups. Baez also dealt with their applications in quantum logic, special relativity, and supersymmetry. Imaeda K and Imaeda M [15] presented a 16-dimensional Cayley–Dickson algebra and found its algebraic properties, zero-divisors, and solutions to a general linear equation. Tian [16] provided a complete investigation of real matrix representations of octonions and considered their various applications to octonions, based on the fact that the octonion is an extension of the quaternion by the Cayley–Dickson construction. Gotô and Nôno [17] established a commutative algebra identified with as the subalgebra of the four-dimensional real matrix algebra . They provided a regularity and properties of the functions of two complex variables with values in . Rudin [18] studied the integral formulas of the holomorphic functions of several complex variables and presented a function theory of including the boundary behavior of complex functions, complex-tangential phenomena, and quantitative theorems about zero-varieties in the unit ball of .

In this paper, we give algebra using the matrix as the basis by mapping the basis of the quaternion to matrix form. We define an algebraic system based on a matrix with a matrix as a component. This can be expressed as a matrix with complex numbers as components while retaining the characteristics of the basis for generating quaternions. Furthermore, a matrix is constructed with a matrix that is isomorphic to the basis of quaternions. The algebra introduced in this paper has eight bases, some of which are commutative for products and some are noncommutative. Therefore, in this paper, a function is defined on an algebra generated by a semicommutative basis. Regularity of functions and properties of these regular functions is investigated. In addition, the corresponding Cauchy–Riemann equations, which can be an alternative of the definition of holomorphic function in complex analysis, are derived for the algebra.

We develop the foundation of a function theory over a subalgebra of four-dimensional matrix algebra to replace the basis of the quaternion. In Section 2, we present preliminaries and notations required for the theory presented in this paper. Sections 3 and 4 present the definitions and propositions related to the semicommutative subalgebra of the complex matrix algebra , and we provide a regularity of functions defined on a domain in with values in . From the notion of regularity over , we propose corresponding Cauchy–Riemann equations and several properties of regular functions in . Finally, in Section 5, the conclusions of this paper are presented.

#### 2. Preliminaries and Notations

In this section, we give notations that are needed to prove our main results. Let be the ring of all matrices over the field . For our purposes, we would be interested in the case where and . Put

These have the following rules:

Let be a set of matrices defined as follows:

From the rules of , we note that is a noncommutative subalgebra of . By using the matrices , we define four matrices:where is the null matrix. Then, we have the following properties:and

Now, let the set be a subalgebra of , written asHere, we deal with as 1. The element of is also written as

From the above expressions, by simply making the following substitutions, we putand we obtainwhere denotes the complex conjugate of the complex components of and is a changing symbol of a part of the basis of . Then, the elements of can be represented in the form . Let be the set , which is the set of elements of expressed in the form of , denoted by

In addition, the conjugate of in is defined as . Explicitly, we have

For , we can write and , then owing to the identities

Thus, we obtain

Since , we have

However,

Although, as a result, , it is equal to a part of and ; hence, is a semicommutative subalgebra of . We first obtain an expression for in order to define . Let , where is the transposed matrix of . Explicitly, (as an element of ) is given by

From the above results, the norm of is therefore given bywhere is the trace of matrix .

Proposition 1. *For , the following properties for the norm are satisfied:*(1)*(2)**We now define differential operators as algebraic expressions for the elemental form of . The differential operator here includes elements in the form of a matrix. We consider the following differential operators over :where has the same pattern operator form as usual complex differential operators.*

As a concrete example, we have

In this case, let the identity matrix be . If the calculation is performed by expressing it as a matrix, the following results are obtained:

Next, let be a domain in . We consider a function bywhere and have the form with and being complex-valued functions. For example, for a function , this takes the form , denoted by and .

The differential operator operates on as follows: in the case of , applying the differential operator from the left, we haveand applying the differential operator from the right, we obtain

Now, let us define the regular functions to be applied in by using the differential operator for functions defined in .

#### 3. Definitions and Properties of Regularity

*Definition 1. *Let be a domain in . Let be a function defined on . The function is said to be -regular function in if(1) and are separately continuously holomorphic functions defined on (2) in , where When a function is -regular , it means that at least the above definition is satisfied for at least one of .

*Remark 1. *Regarding and being separately continuous holomorphic functions, we mean that and such that the components and of each are holomorphic complex-valued functions.

Lemma 1 (Cauchy–Riemann equation in ). *Condition 2 of Definition 1 for -regular function is equivalent to**In addition, for -regular function, Condition 2 is equivalent to*

Theorem 1. *Let be an open set in , and let be a regular function on . Then, the derivative of , also denoted as , is defined ason .*

*Proof. *From the definition of -regular function defined on , in case , Condition 2 can be expressed as follows:Since satisfieswe obtainSimilarly, we can derive the Cauchy–Riemann equations for the other cases.

For example, let be an identity function such that . The function is -regular defined on . Let be a function such that be -regular defined on . Let be -regular functions defined in . Then, we propose the -derivative of and find the following theorem.

Theorem 2. *Let be a domain in . Let be a -regular function defined in . Then, the derivative of satisfies*

*Proof. *Since is -regular defined on , the function satisfies . Hence, we haveWhen , by using the equation , we obtain the same results, that is,Similarly, when is -regular defined on , then by the equation , we obtainThus, the -derivative of satisfiesFrom Theorem 1, for a separately continuous holomorphic -regular function defined in , we denote

#### 4. Properties of -Regular ( Functions

In this section, let be a domain in . Based on the definition and properties of -regularity , we determine whether the properties of holomorphic function of complex variables extend -regular function defined on . Suppose that are two open neighborhoods; then, the variables in can be written as and can be written as . Any mapping can be described by

The mapping is called a -regular mapping if the functions and are -regular functions in . If is any function defined in , then composition is a well-defined function in . If is a -regular function in and if is a -regular mapping , then the composition is a -regular function in .

Next, the following quadratic form is factors that play an important role in the integral calculus of -regular functions :

Furthermore, we note that

*Definition 2 (Morera’s theorem in ). *Let a function be continuous in . Suppose the integral of over the boundary of any sphere in be equal to zero. Then, is a -regular function in .

*Proof. *We assume that the function is smooth in . Let be a sphere in . Based on this hypothesis, we haveFrom Stokes’ formula in complex analysis, we obtainHence, for , the integral can be calculated as follows:Due to the smoothness of , the expression under the integral sign is continuous. Since the fact is an arbitrary sphere in , it follows thatin . For cases , we can obtain similar conclusions.

Next, we consider the relationship between the -regular function and series expansions in .

Lemma 2. *Let be a homogeneous polynomial of degree with respect to and . If is a -regular function in , then satisfies*

*Proof. *According to the hypothesis, the function is a homogeneous polynomial; thus, satisfiesSince is a homogeneous polynomial of degree , we haveContinuing this process, we can obtain

Proposition 2. *Let be a domain in and be a function defined on a neighborhood of with values in . The function is a -regular function in and if and only if has a power series expansion in such that for ,where .*

*Proof. *Without loss of generality, let . Suppose that is -regular in with a value in ; then, is holomorphic in . Thus, there exists a neighborhood of zero such that for ,where are homogeneous polynomials of degree with respect to and . Since the above series converges uniformly in and are homogeneous polynomials of degree with respect to and , when , we haveHence, is a -regular in . By Lemma 2, sincewe haveSuppose thatwhere . Since converges uniformly in , we obtainThus, is a -regular function in .

From Proposition 2, equivalently, we can define -regular functions in which is a domain in as follows.

*Definition 3. *Let be a domain in . A function is said to a -regular function on if every point corresponds to a neighborhood admitting a power serieswhich converges to for .

From the above results, the following statements are equivalent to each other:(1) is a -regular function defined on .(2)For every point , there is a neighborhood admitting a power series(3)For a smooth boundary such that , we have where is the quaternion product of the form of the function .

Corollary 1. *Let be a domain in and be a -regular function . If is infinitely differentiable in andthen we have*

#### 5. Conclusion

In this paper, we developed the foundation of a function theory over a subalgebra of four-dimensional matrix algebra as an alternative to the standard basis of quaternion algebra. Furthermore, we introduced a semicommutative subalgebra of the complex matrix algebra , and we propose regular functions defined in a domain in with values in . From the definition of regularity over , we derived corresponding Cauchy–Riemann equations and several properties of regular functions in . Throughout this paper, it has been possible to make various attempts to utilize quaternions by demonstrating a series expansion of quaternion variable. Regular functions defined on quaternions can be applied to this extension of the algebra of quaternions introduced in this paper. In addition, it is possible to present the extensibility of formulas derived in this paper based on proposed quaternionic theory for differential operators and series expansion. In the future, we plan to verify properties, such as -closed form and regularity over a domain of holomorphy, by using certain regular functions and Cauchy–Riemann equations.

#### Data Availability

No datasets were generated or analyzed during the current study.

#### Conflicts of Interest

The author declares no conflicts of interest.

#### Acknowledgments

This study was supported by the Dongguk University Research Fund 2020 and the National Research Foundation of Korea (NRF) (2021R1F1A1063356).