Quaternion Rhapsody

When I started my research in area of calculus over division ring, I paid attention to large volume of papers dedicated to regular functions of quaternion and analogue of the Cauchy-Riemann equation. I wanted to understand whether the Cauchy-Riemann equation appears in the frame of the theory that I explore. This was the reason why I considered division ring as vector space over center.

Introduction of basis simplifies some constructions and presents a bridge between the Gâteaux derivative and Jacobian matrix of map. This is exactly the place, where the Cauchy-Riemann equations should appear. The exploration of derivative of function of complex numbers reveals that the Cauchy-Riemann equation has algebraic origin and is related with statement that there exists $R$-linear function over complex field, however this function is not $C$-linear. For instance, conjugation of complex number is linear over real field, however it is not linear map over complex field.

In quaternion algebra $H$, there exist only $R$-linear map. The corollary of this statement is the ability represent every $R$-linear map using quaternion and absence of the evident analogue of the Cauchy-Riemann equation in quaternion algebra. However, in numerous papers and books dedicated to calculus over quaternion algebra mathematicians explore sets of maps that have properties similar to properties of functions of complex variable. Some of authors do not restrict themselves to quaternion algebra and explore more general algebras.

In the paper [3], Gelfand explores the quaternion algebra over arbitrary field assuming that product depends on arbitrary parameters. We assume $H=E(R,-1,-1)$. Using this paper, I decided to explore two cases that are important for me.

The algebra $E(R,a,b)$ was interesting for me because I supposed to find parameters $a$, $b$ such that the system of linear equations [2]-(LABEL:0812.4763-English-eq:_linear_map_over_field,_division_ring,_relation) is singular. It was important to understand what happens in this case. When I explored the structure of linear map over division ring it was not evident how non singularity of system of linear equations [2]-(LABEL:0812.4763-English-eq:_linear_map_over_field,_division_ring,_relation) have an influence on answer. However, the solution to this problem was not the one I had expected. It turned out that the system of linear equations is so simple that everybody can see that this system cannot be singular.

I have wrote that the Cauchy-Riemann equation is related with statement that complex field has real field as subfield. I assumed also that similar statement is possible in algebras with enough aggregate center. This is why I expected to see analogue of the Cauchy-Riemann equation in the quaternion algebra over complex field. In the course of solving the problem I realized that algebra $E(C,-1,-1)$ is isomorphic to tensor product $C\otimes H$. Therefore linear functions of this algebra satisfy to the Cauchy-Riemann equation for $C$-component of tensor product. Therefore I can tell the same about Jacobian matrix of arbitrary function. Natural extension of this topic is exploration of tensor product $C\otimes(C\otimes H)$. In particular, I put my attention is non associativity of tensor product.

In the book [2] on the base of which I wrote this paper, I explore the Gâteaux derivative of function over division ring. However in this paper I consider arbitrary algebras that not always are division rings. During the time that I explore the Gâteaux derivative, I realized that this subject can be generalized to more wide set of algebras. I will prepare the complete research later. I wrote this paper in order to explore conditions when the Cauchy-Riemann equations are possible.

1. (1)

   Function and mapping are synonyms. However according to tradition, correspondence between either rings or vector spaces is called mapping and a mapping of either real field or quaternion algebra is called function.

2. (2)

   We can consider division ring $D$ as $D$-vector space of dimension $1$. According to this statement, we can explore not only homomorphisms of division ring $D_{1}$ into division ring $D_{2}$, but also linear maps of division rings. This means that map is multiplicative over maximum possible field. In particular, linear map of division ring $D$ is multiplicative over center $Z(D)$. This statement does not contradict with definition of linear map of field because for field $F$ is true $Z(F)=F$. When field $F$ is different from maximum possible, I explicit tell about this in text.

3. (3)

   Let $A$ be free finite dimensional algebra. Considering expansion of element of algebra $A$ relative basis $\overline{\overline{e}}{}$ we use the same root letter to denote this element and its coordinates. However we do not use vector notation in algebra. In expression $a^{2}$, it is not clear whether this is component of expansion of element $a$ relative basis, or this is operation $a^{2}=aa$. To make text clearer we use separate color for index of element of algebra. For instance,

   $$a=a^{\boldsymbol{{\color[rgb]{1,.3,.6}i}}}\overline{e}{}_{\boldsymbol{{\color[rgb]{1,.3,.6}i}}}$$

4. (4)

   If free finite dimensional algebra has unit, then we identify the vector of basis $\overline{e}{}_{\boldsymbol{{\color[rgb]{1,.3,.6}0}}}$ with unit of algebra.

5. (5)

   Without a doubt, the reader may have questions, comments, objections. I will appreciate any response.

Equation (3.11) is equivalent to equation

In this paper I explore the set of quaternion algebras defined in [3].

Elements of the algebra $E(F,a,b)$ have form

where $x^{\boldsymbol{{\color[rgb]{1,.3,.6}i}}}\in F$, $\boldsymbol{{\color[rgb]{1,.3,.6}i}}=\boldsymbol{{\color[rgb]{1,.3,.6}0}}$, $\boldsymbol{{\color[rgb]{1,.3,.6}1}}$, $\boldsymbol{{\color[rgb]{1,.3,.6}2}}$, $\boldsymbol{{\color[rgb]{1,.3,.6}3}}$. Quaternion

is called conjugate to the quaternion $x$. We define the norm of the quaternion $x$ using equation

From equation (4.2), it follows that $E(F,a,b)$ is algebra with division only when $a<0$, $b<0$. In this case we can renorm basis such that $a=-1$, $b=-1$.

We use symbol ${\color[rgb]{.4,0,.9}E(F)}$ to denote the quaternion division algebra $E(F,-1,-1)$ over the field $F$. We will use notation ${\color[rgb]{.4,0,.9}H}=E(R,-1,-1)$. Multiplication in quaternion algebra $E(F)$ is defined according to rule

In algebra $E(F)$, the norm of the quaternion has form

In this case inverse element has form

The inner automorphism of quaternion algebra $H$²²2See [6], p.643.

describes the rotation of the vector with coordinates $x$, $y$, $z$. The norm of quaternion $q$ is irrelevant, although usually we assume $|q|=1$. If $q$ is written as sum of scalar and vector

then (4.6) is a rotation of the vector $(x,y,z)$ about the vector $(a,b,c)$ through an angle $2\alpha$.

Let $F_{1}$ be algebra over the field $F_{2}$. Let $\overline{\overline{e}}{}_{12}$ be basis of algebra $F_{1}$ over the field $F_{2}$. Let $C_{12\cdot}{}_{\boldsymbol{{\color[rgb]{1,.3,.6}ij}}}^{\boldsymbol{{\color[rgb]{1,.3,.6}k}}}$ be structural constants of algebra $F_{1}$ over the field $F_{2}$.

Let $F_{2}$ be algebra over the field $F_{3}$. Let $\overline{\overline{e}}{}_{23}$ be basis of algebra $F_{2}$ over the field $F_{3}$. Let $C_{23\cdot}{}_{\boldsymbol{{\color[rgb]{1,.3,.6}ij}}}^{\boldsymbol{{\color[rgb]{1,.3,.6}k}}}$ be structural constants of algebra $F_{2}$ over the field $F_{3}$.

I will consider the algebra $F_{1}$ as direct sum of algebras $F_{2}$. Each item of sum I identify with vector of basis $\overline{\overline{e}}{}_{12}$ . Accordingly, I can consider algebra $F_{1}$ as algebra over field $F_{3}$. Let $\overline{\overline{e}}{}_{13}$ be basis of algebra $F_{1}$ over the field $F_{3}$. Index of basis $\overline{\overline{e}}{}_{13}$ consists from two indexes: index of fiber and index of vector of basis $\overline{\overline{e}}{}_{23}$ in fiber.

I will identify vector of basis $\overline{e}{}_{12\cdot\boldsymbol{{\color[rgb]{1,.3,.6}i}}}$ with unit in corresponding fiber. Then

The product of vectors of basis $\overline{\overline{e}}{}_{13}$ has form

Because $C_{12\cdot}{}_{\boldsymbol{{\color[rgb]{1,.3,.6}ik}}}^{\boldsymbol{{\color[rgb]{1,.3,.6}b}}}\in F_{2}$, then expansion $C_{12\cdot}{}_{\boldsymbol{{\color[rgb]{1,.3,.6}ik}}}^{\boldsymbol{{\color[rgb]{1,.3,.6}b}}}$ relative to basis $\overline{\overline{e}}{}_{23}$ has form

Let us substitute (5.3) into (5.2)

Therefore, we can define structural constants of algebra $F_{1}$ over field $F_{3}$

To verify construction, let us consider the product

On the other hand

Expressions (5.6), (5.7) coincide.

I intentionally wrote the equation (5.22) in the intermediate form, to show how much more is tedious proof of the theorem 5.5 compared with proof of the analogous theorem 3.1. never used the statement that the product in the $F_{1}$-algebra $F_{2}$ is commutative. Therefore, the theorem 5.4 can be applied to an arbitrary algebra.

In this section, I will consider quaternion algebra $E(C,-1,-1)$, where $C$ is complex field.

Product in algebra $E(C,-1,-1)$ is defined according to table