On Structure of Octonion Regular Functions

In this paper we study octonion regular functions and the structural differences between regular functions in octonion, quaternion, and Clifford analyses.


Introduction
In our recent papers [5,6], we started to study octonion algebraic methods in analysis. This paper is a continuation of our studies in this fascinating field. Over the years, many results of octonion analysis have been published and studied since the fundamental paper of Dentoni and Sce [2]. One thing which has remained unclear to us is that what is octonion analysis all about? A consensus has been that octonion, quaternion and Clifford analyses are similar from a theoretical point of view, and maybe for this reason octonion analysis has been left to less attention. Our aim is to prove that octonion analysis and Clifford analysis are different theories from the point of view of regular functions. Thus, octonion analysis is a completely independent research topic.
We start by recalling preliminaries of octonions and Clifford numbers and their connections via triality. We define our fundamental function classes, i.e., left-, right-and bi-regular functions. We give chararacterizations for function classes in biaxial quaternion analysis and in Clifford analysis. The classical Riesz system of Stein and Weiss is used as a familiar reference to clearly see the differences.
The topic of this paper is highly technical, but we have tried to write everything in as a simple way as possible. Hopefully we have succeeded in this job. Many questions remain open and the reader may find a lot of open research problems between the lines. Hopefully we can answer some of these questions when the saga continues.

Preliminaries: Octonion and Clifford algebras
In this algebraic part of the paper, we first recall briefly the basic definitions and notations related to the octonion and Clifford algebras. Then we study their connections in detail. In the whole paper, our principle is to consider the standard orthonormal basis {e 0 , e 1 , . . . , e 7 } for R 8 , and by defining different products between the elements, we obtain different algebras. We will denote the octonion product by e i • e j , and the Clifford product by e i e j .

Octonions
The algebra of octonions O is the non-commutative and non-associative 8-dimensional algebra with the basis {1, e 1 , . . . , e 7 } and the multiplication given by the following table. Let us point out that there are 480 possible ways to define an octonion product such that e 0 = 1.
Our choise is historically maybe the most used and traditional, and for this reason we may call it the canonical one, but e.g. Lounesto uses a different definition for octonion multiplication in his famous book [8].
An element x ∈ O may be represented in the forms Here, x 0 , ..., x 7 ∈ R, x 0 is the real part, x is the vector part, and u and v ∈ H are quaternions. The last form is called the quaternion form of an octonion. The conjugate of x is denoted and defined by x = x 0 − x. We see that the element e 4 plays a kind of a role of the "imaginary unit". The product of two octonions can be written as where x · y is the dot product and x × y the cross product of vectors x and y.
Denote the quaternion forms of octonions x and y by In Lemma 2.3 we will return the cross product x × y of octonion vector parts x and y to cross products of the vector parts u, v, a, and b of quaternions, which are classical 3-dimensional cross products (see, e.g., [3,8,10]).
Lemma 2.2. If x, y ∈ O be as in (2), then Lemma 2.3. Denote the quaternion representations of the vectors x and y ∈ O as in (2). Then the cross product in quaternion form is Proof. By Lemma 2.2, we compute Corollary 2.4. If x, y ∈ O be as in (2), then • e 4 ∈ span{e 5 , e 6 , e 7 } 2.2 The Clifford algebra Cℓ 0,7 and triality Since the dimension of octonions and Clifford paravectors is 8, they behave similarly as vector spaces. Moreover, we may ask if there is a connection between the octonion and the Clifford product? The answer is given by Pertti Lounesto in his book [8]. We will recall his ideas here in detail. Let us recall the basic definitions and properties of Clifford algebras.
We continue working with the basis {e 0 , e 1 , ..., e 7 } for R 8 . The Clifford product is defined by where δ ij is the Kronecker delta symbol. Here, e 0 = 1. Then, similarly than in the case of octonions, e 2 0 = 1, and e 2 j = −1 for all j = 1, . . . , 7. The Clifford product e i e j is not necessary a vector or a scalar. This product generates an associative algebra, called the Clifford algebra, denoted by Cℓ 0,7 . The dimension of this Clifford algebra is 2 7 , and an element a ∈ Cℓ 0,7 may be represented as a sum a = The Clifford product of two paravectors x and y can be written where x ∧ y is the wedge product of vectors x and y. In particular, xy = x ∧ y − x · y.
The reader can see that formally the octonion and the Clifford products are similar, and a reasonable question is, how they are connected? We would like to construct the octonion product using the Clifford algebra Cℓ 0,7 . Let us consider the octonion product of the basis elements e i and e j , Then 1 ≤ k ≤ 7, and i = k = j. The corresponding Clifford product e i e j may be mapped to e k by multiplying it by the trivector e j e i e k , i.e., and by the same trivector e j e i is mapped to −e k . If a and b are vectors, then 5 .
gives us a part of the kth component of the octonion product a • b. Using this idea, we may express the octonion product a • b as the paravector part of the Clifford product ab(1 − W ), where W is a suitable 3-vector. Let a = a 0 + a and b = b 0 + b be paravectors. Then Lounesto states Lemmas 2.5 and 2.7 without proofs at pages 303-304 in [8], and for a different multiplication table of octonions. Venäläinen gives a proof for Lemma 2.5 in her licentiate thesis [12]. For the convenience of the reader, we give a short proof of Lemma 2.5 here.
Proof. We compute By (1) and (3), it is enough to show that The product e i e j e ν is a vector only if the pair of indices ij belongs to the triplet ν. Since the cross and the wedge products have the same coefficients, and each pair ij, 1 ≤ i < j ≤ 7, is contained in exatly one of the triplets ν, say ν 0 , it is enough to check that e i e j e ν0 = −e i • e j for all such pairs ij.
A straightforward computation shows that the trivector W is invertible.
In the above, we identify octonions O with the 8-dimensional paravectors. The dimension 8 plays a special role in the theory of spin groups, since Spin (8) has the so called exceptional automorphisms. This feature is called triality, and the first time it was noticed in the book of Study [11]. For modern references, see [4,8,10]. The triality means that in addition to paravectors, we may identify octonions with the spinor spaces S ± . The spinor spaces may be realized by where I ± is a primitive idempotent, see [1,8]. A straightforward computation shows that the octonion product in spinor spaces may be computed as follows.

Octonion analysis
In this section we recall the basic facts of octonion analysis, i.e., the theory of Cauchy-Riemann operators in the octonionic setting. After that, we carefully study the general structure of the null solutions of these operators and define three different classes of regular functions: left-, right-, B-, and R-regular functions. R-regular functions are just solutions of the classical Riesz system. We use the Riesz system here as a familiar reference to better understand the structure of octonion regular functions. In Clifford analysis the corresponding function classes are equal. This structural difference is a fundamental difference between octonion and Clifford analyses.

Cauchy-Riemann operators
We define the Cauchy-Riemann operator The vector part of it is called the Dirac operator. If the coordinate functions of f have partial derivatives, then D x operates on f from the left and from the right as Decomposition (1) gives (see [7]) where If D x f = 0 (resp. f D x = 0), then f is called left (resp. right ) regular.
In Clifford analysis one studies functions f : R 8 → Cℓ 0,7 . We define the Cauchy-Riemann operator similarly as in octonion analysis: Functions satisfying ∂ x f = 0 (resp. f ∂ x = 0) on R 8 are called left (resp. right ) monogenic. In this paper we only need to consider paravector valued functions

Left-, Right-, B-and R-regular functions
Comparing the real and vector parts in (6) and (7) yields the following well known results.
or componentwise We will denote the space of left regular functions by M (ℓ) , and similary, right regular functions by M (r) .

Proposition 3.2. A function f : R 8 → O is both left and right regular if and only if it satisfies the system
or componentwise We will call functions satisfying (11) B-regular, and denote the space of such functions by M B . Naturally The fundamental difference between octonion and Clifford analyses is that in Clifford analysis the paravector valued null solutions to the Cauchy-Riemann operator satisfies the Riesz system and are at the same time left and right monogenic, which is not true in octonion analysis. The following well-known proposition follows from the definitions similarly as in octonion analysis by comparing the scalar parts, 1-vector parts, and 2-vector parts.
or componentwise Functions satisfying (13) are called R-regular, and the space of such functions is denoted by M R .
To convince the reader about the existence of these function classes, we recall the following classical method from Clifford analysis. We may conclude that although Clifford and octonion analyses have formally very similar definitions, the corresponding function spaces are different.
This result is crucial in understanding the fundamental character of octonion analysis and the structural differences between octonion, quaternion, and Clifford analyses.

Characterizing function classes in biaxial quaternion analysis
In the preceding section we gave characterizations for left-, B-, and R-regular functions using componentwise and vector forms. In this section we write the three systems of Section 3.2 in quaternion forms. The use of the quaternion forms of the function and the Cauchy-Riemann operator is called the biaxial quaternion analysis.
Write the Cauchy-Riemann operator D x and the function f : R 8 → O in the quaternion forms: According to Corollary 2.4, we can write This implies the quaternion forms of the Moisil-Teodorescu type system (8) and the system (10). (14) One example of the use of the biaxial quaternion analysis is the proof of the following vector calculus identity in the octonionic case.
Proof. We use quaternion decompositions On the left-hand side we apply Lemma 2.3 to the cross product f × g, and use the classical vector calculus identity On the right-hand side we apply Lemma 2.3 to the rotors D x × f and D x × g:  Then and Here, Corollary 4.6. Let the coordinates of f and g : H → H have partial derivatives. Then Proof. Apply Lemmas 2.1 and 4.5, and use the fact f g = g f .

Characterization of the function classes in Clifford analysis
In this last section, we study the classes of left-, B-, and R-regular functions using Clifford analysis. We begin with the following algebraic lemma.
To prove the last claim, it is now enough to show that in the case [abI] 0 = 0, [abI] 2 = 0 if and only if [abI] 3 = 0. This can be seen by computing and in the case [abI] 0 = 0, We infer that left-, B-, and R-regularity can be studied by considering paravector-spinor valued functions f I. (b) We compute, using (3) and (26), Since D x × f = −[(∂ x ∧ f )W ] 1 (Lemma 2.5), the claim now follows from (a) and Propositions 3.1-3.2.

Conclusion
The idea of this paper is to study differences between octonion and Clifford analyses. This leads us to observe the fundamental difference between octonion regular and Clifford monogenic functions. The structure of octonion regular functions is studied by comparing left-, right-, B-, and R-regular functions. The existence of these classes is a consequence of different algebraic properties of the algebras. In the heart of octonion analysis is the study of the properties of these function classes and their relations, which distinguishes it essentially from Clifford analysis.