New Characterizations of Algebraic Regularity

TL;DR

This paper introduces new characterizations of algebraic regularity utilizing differential forms and difference quotients, providing fresh perspectives and tools for understanding algebraic structures.

Contribution

It offers novel characterizations of algebraic regularity through the use of differential forms and difference quotients, expanding theoretical understanding.

Findings

New characterizations of algebraic regularity established

Differential forms and difference quotients as key tools

Enhanced theoretical framework for algebraic structures

Abstract

In this paper, we give new characterizations of algebraic regularity by using differential forms and difference quotients.

Equations

$$e_2^2=e_3^2=e_4^2=-e_1,e_i{e}_j=-e_j{e}_i=(-1{)}^{i+j+1}{e}_{9-i-j},$$

$$f(x)=f_1(x_1,x_2,x_3,x_4)e_1+\sum _{k=2}^4{x}_k{f}_0(x_1,x_2,x_3,x_4)e_k$$

$$\frac{\partial f_1}{\partial x_1}=f_0+x_2\frac{\partial f_0}{\partial x_2}+x_3\frac{\partial f_0}{\partial x_3}+x_4\frac{\partial f_0}{\partial x_4},$$

$$\frac{\partial f_1}{\partial x_i}=-x_i\frac{\partial f_0}{\partial x_1},x_i\frac{\partial f_0}{\partial x_j}=x_j\frac{\partial f_0}{\partial x_i},$$

$$f(x)=f\left(\sum _{k=1}^4{x}_i{e}_k\right)=\sum _{k=1}^4{f}_k(x_1,x_2,x_3,x_4)e_k,$$

$$f_k(x_1,x_2,x_3,x_4)=f_k(x)=f_k(x_1{e}_1+x_i{e}_i+x_j{e}_j+x_{9-i-j}{e}_{9-i-j}),$$

$$(d{x}^i)(q_1{e}_1+q_2{e}_2+q_3{e}_3+q_4{e}_4):=q_i,$$

$$\alpha =\sum _{1\le i_1,\dots ,i_m\le 4}{f}_{i_1,\dots ,i_m}(x)d{x}^{i_1}\wedge d{x}^{i_2}\wedge \cdots \wedge d{x}^{i_m},$$

$$\beta =\sum _{1\le j_1,\dots ,j_n\le 4}{g}_{j_1,\dots ,j_n}(x)d{x}^{j_1}\wedge d{x}^{j_2}\wedge \cdots \wedge d{x}^{j_n}$$

$$\displaystyle\sum_{\begin{array}[]{c}1\leq i_{1},\dots,i_{m}\leq 4\\ 1\leq j_{1},\dots,j_{n}\leq 4\end{array}}f_{i_{1},\dots,i_{m}}(x)g_{j_{1},\dots,j_{n}}(x)\,dx^{i_{1}}\wedge\cdots\wedge dx^{i_{m}}\wedge dx^{j_{1}}\wedge\cdots\wedge dx^{j_{n}}.$$

$$df=\frac{\partial f}{\partial x_1}d{x}^1+\frac{\partial f}{\partial x_2}d{x}^2+\frac{\partial f}{\partial x_3}d{x}^3+\frac{\partial f}{\partial x_4}d{x}^4,$$

$$d(f+g)=df+dg,$$

$$d(fg)=f\wedge (dg)+(df)\wedge g,$$

$$d(qf)=q\,(df),\qquad d(fq)=(df)\,q\qquad\mbox{for $q\in\mathcal{H}$.}$$

$$d\alpha :=\sum _{1\le i_1,\dots ,i_m\le 4}d(f_{i_1,\dots ,i_m})\wedge d{x}^{i_1}\wedge d{x}^{i_2}\wedge \cdots \wedge d{x}^{i_m}.$$

$$Dq:=e_{1}d{x}^2\wedge d{x}^3\wedge d{x}^4-e_{2}d{x}^1\wedge d{x}^3\wedge d{x}^4+e_{3}d{x}^1\wedge d{x}^2\wedge d{x}^4-e_{4}d{x}^1\wedge d{x}^2\wedge d{x}^3$$

$$Dq:=e_{1}d{x}^2\wedge d{x}^3\wedge d{x}^4+\sum _{2\le i<j\le 4}(-1{)}^{i+j}{e}_{9-i-j}d{x}^1\wedge d{x}^i\wedge d{x}^j.$$

$${\mathcal{D}}_{\ell }(f):$$

$${\mathcal{D}}_r(f):$$

$$\frac{\partial f}{\partial x_{i}}:=\Big{(}\frac{\partial f_{1}}{\partial x_{i}}\Big{)}e_{1}+\Big{(}\frac{\partial f_{2}}{\partial x_{i}}\Big{)}e_{2}+\Big{(}\frac{\partial f_{3}}{\partial x_{i}}\Big{)}e_{3}+\Big{(}\frac{\partial f_{4}}{\partial x_{i}}\Big{)}e_{4}\quad\mbox{for $1\leq i\leq 4$}.$$

$$Dq\wedge df=-{\mathcal{D}}_{\ell }(f)v\text{mbox}anddf\wedge Dq={\mathcal{D}}_r(f)v,$$

$$D_{0}q:=(x_2+x_3+x_4)d{x}^2\wedge d{x}^3\wedge d{x}^4-\sum _{2\le i<j\le 4}(-1{)}^{i+j}{x}_{9-i-j}d{x}^1\wedge d{x}^i\wedge d{x}^j;$$

$$D_{1}q:=d{x}^2\wedge d{x}^3\wedge d{x}^4+\sum _{2\le i<j\le 4}(-1{)}^{i+j}d{x}^1\wedge d{x}^i\wedge d{x}^j;$$

$$D_{0}q\wedge df=-df\wedge D_{0}q$$

$$D_{1}\,q\wedge df=-df\wedge D_{1}\,q=-\Big{(}\frac{\partial f}{\partial x_{1}}+\frac{\partial f}{\partial x_{2}}+\frac{\partial f}{\partial x_{3}}+\frac{\partial f}{\partial x_{4}}\Big{)}\,v,$$

$$f(x)=f_1(x_1,x_2,x_3,x_4)e_1+\sum _{k=2}^4{x}_k{f}_0(x_1,x_2,x_3,x_4)e_k,$$

$$Dq\wedge df+2D_{0}q\wedge d{f}_0+2D_{1}q\wedge d{f}_1=0$$

$$df\wedge Dq+2d{f}_0\wedge D_{0}q+2d{f}_1\wedge D_{1}q=0$$

$${\overleftarrow{f}}_{9-i-j}^c(x):$$

$${\overrightarrow{f}}_{9-i-j}^c(x):$$

$$f(x)=f_1(x_1,x_2,x_3,x_4)e_1+\sum _{i=2}^4{x}_i{f}_0(x_1,x_2,x_3,x_4)e_i,$$

Full text

1 Introduction

In [2], we introduced the algebraic regularity on the quaternions by using a new generalization of Cauchy-Riemann system, and characterized the new generalization of Cauchy-Riemann system by using Fueter-operators. In this paper, we characterize the algebraic regularity on the quaternions by using differential forms and difference quotients.

Throughout this paper, we let $\mathcal{H}=\mathcal{R}e_{1}\oplus\mathcal{R}e_{2}\oplus\mathcal{R}e_{3}\oplus\mathcal{R}e_{4}$ be the quaternions discovered by W. R. Hamilton in 1843, where $\mathcal{R}$ is the real number field, $e_{1}$ is the identity of the real division associative algebra $\mathcal{H}$, and the multiplication among the remaining three elements in the $\mathcal{R}$-basis $\{e_{1},\,e_{2},\,e_{3},\,e_{4}\}$ is defined by

[TABLE]

where $2\leq i<j\leq 4$.

Recall that the algebraic regularity on the quaternions is defined in the following way:

Definition 1.1

Let $U$ be an open subset of $\mathcal{H}$. We say that a quaternion-valued function $f:U\to\mathcal{H}$ is algebraic regular at $c=\displaystyle\sum_{i=1}^{4}c_{i}e_{i}\in U$ if $f$ has two properties given below.

(i)

There exist two $C^{1}$ real-valued functions $f_{0}:\mathcal{R}^{4}\to\mathcal{R}$ and $f_{1}:\mathcal{R}^{4}\to\mathcal{R}$ such that

[TABLE]

for all $x=\displaystyle\sum_{i=1}^{4}x_{i}e_{i}\in U$.

(ii)

The following equations hold at $(c_{1},\,c_{2},\,c_{3},\,c_{4})\in\mathcal{R}^{4}$:

[TABLE]

[TABLE]

where $2\leq i,\,j\leq 4$ and $i\neq j$.

We say that $f:U\to\mathcal{H}$ is an algebraic regular function on $U$ if $f$ is algebraic regular at every point of $U$.

2 Characterizing Algebraic Regularity by Differential Forms

Let $U$ be an open subset of the quaternion $\mathcal{H}$. A quaternion-valued function $f:U\to\mathcal{H}$ has the form

[TABLE]

where $x=\displaystyle\sum_{k=1}^{4}x_{k}e_{k}\in U$, $x_{1},\,x_{2},\,x_{3},\,x_{4}\in\mathcal{R}$ and $f_{k}(x_{1},\,x_{2},\,x_{3},\,x_{4})$ is a real-valued function of four real variables $x_{1}$, $x_{2}$, $x_{3}$ and $x_{4}$. The quaternion-valued function $f:U\to\mathcal{H}$ given by (4) is said to be smooth ( or $C^{1}$) if the real-valued function $f_{k}(x_{1},\,x_{2},\,x_{3},\,x_{4})$ is smooth (or $C^{1}$) for $1\leq k\leq 4$. The alternate notations for the real-value function $f_{k}(x_{1},\,x_{2},\,x_{3},\,x_{4})$ are given as follows

[TABLE]

where $x=\displaystyle\sum_{k=1}^{4}x_{k}e_{k}$ and $2\leq i\neq j\leq 4$.

For $1\leq i\leq 4$, we define $dx^{i}\in Hom_{\mathcal{R}}(\mathcal{H},\,\mathcal{R})$ by

[TABLE]

where $q_{i}\in\mathcal{R}$ for $1\leq i\leq 4$. Clearly, $\{dx^{1},\,dx^{2},\,dx^{3},\,dx^{4}\}$ is a basis for the real vector space $Hom_{\mathcal{R}}(\mathcal{H},\,\mathcal{R})$.

A quaternion-valued $m$-form on an open subset $U$ of $\mathcal{H}$ is an expression of the form

[TABLE]

where $f_{i_{1},\dots,i_{m}}(x):U\to\mathcal{H}$ is a smooth quaternion-valued function on $U$, and $dx^{i_{1}}\wedge dx^{i_{2}}\wedge\cdots\wedge dx^{i_{m}}$ is the ordinary alternating $\mathcal{R}$-multilinear map from $\underbrace{\mathcal{H}\times\cdots\times\mathcal{H}}_{m}$ to $\mathcal{R}$. By (5), a quaternion-valued [math]-form on an open subset $U$ of $\mathcal{H}$ is a quaternion-valued function defined on $U$. The smooth quaternion-valued functions $f_{i_{1},\dots,i_{m}}(x)$ are called the coefficients of $\alpha$. We say that $\alpha$ is an real-valued $m$ form if all of its coefficients are smooth real-valued functions.

The exterior product of a quaternion-valued $m$-form $\alpha$ given by (5) and a quaternion-valued $n$-form $\beta$

[TABLE]

is defined to be the quaternion-valued $(m+n)$-form $\alpha\wedge\beta$ which is given by :

[TABLE]

Proposition 2.1

Let $\alpha$ and $\beta$ be quaternion-valued $m$-form $\alpha$ given by (5) and a quaternion-valued $n$-form given by (6). If either $\alpha$ or $\beta$ is a real-valud form, then $\alpha\wedge\beta=(-1)^{mn}\beta\wedge\alpha$.

If $f:U\to U$ is a quaternion-valued [math]-form given by $f(x)=\displaystyle\sum_{i=1}^{4}f_{i}(x)\,e_{i}$ with the real-valued functions $f_{1}(x)$, $f_{2}(x)$,$f_{3}(x)$ and $f_{4}(x)$, we define the **differential ** $df$ of $f$ to be the quaternion-valued $1$-form on $U$ given by

[TABLE]

where $\displaystyle\frac{\partial f}{\partial x_{i}}:=\displaystyle\sum_{j=1}^{4}\displaystyle\frac{\partial f_{j}}{\partial x_{i}}\,e_{j}$ is a quaternion-valued function for $1\leq i\leq 4$.

Proposition 2.2

If $f$ and $g$ are quaternion-valued [math]-forms on an open subset $U$ of $\mathcal{H}$, then

[TABLE]

[TABLE]

[TABLE]

If $\alpha$ is a quaternion-valued $m$-form given by (5), we define $d\alpha$ is the quaternion-valued $(m+1)$-form given by

[TABLE]

The operator $d$ is called the exterior differentiation. The next proposition gives the basic properties of the exterior differentiation.

Proposition 2.3

(i)

If $q\in\mathcal{H}$ and $\alpha$, $\beta$ are quaternion-valued $m$-forms, then $d(\alpha+\beta)=d(\alpha)+d(\beta)$, $d(q\alpha)=q\,d(\alpha)$ and $d(\alpha q)=d(\alpha)\,q$.

(ii)

If $\alpha$ is a quaternion-valued $m$-form and $\beta$ is a quaternion-valued $n$-form, then $d(\alpha\wedge\beta)=(d\alpha)\wedge\beta+(-1)^{m}\alpha\wedge(d\beta)$.

(iii)

If $\alpha$ is a quaternion-valued $m$-form, then $d(d\alpha)=0$.

Following [3], we use $D\,q$ to denote the following quaternion-valued $3$-form:

[TABLE]

or

[TABLE]

Also, recall from [3] that left Fueter operator operators \mathcal{D}_{\ell}:=\displaystyle\sum_{i=1}^{4}e_{i}\Big{(}\frac{\partial}{\partial x_{i}}\Big{)} and the right Fueter operator \mathcal{D}_{r}:=\displaystyle\sum_{i=1}^{4}\Big{(}\frac{\partial}{\partial x_{i}}\Big{)}e_{i} are defined by

[TABLE]

where

[TABLE]

Proposition 2.4

If $f:U\to\mathcal{H}$ is a $\mathcal{C}^{1}$ quaternion-valued function defined on an open subset of $\mathcal{H}$, then

[TABLE]

where $v=dx^{1}\wedge dx^{2}\wedge dx^{3}\wedge dx^{4}$ is the volume form.

We now introduce two more real-valued $3$-forms $D_{0}\,q$ and $D_{1}\,q$ as follows:

[TABLE]

[TABLE]

The basic properties of the two real-valued $3$-forms above are given in the following

Proposition 2.5

Let $U$ be an open subset of $\mathcal{H}$. If $f$ is a $\mathcal{C}^{1}$ quaternion-valued function defined on $U$, then the following equations hold on $U$:

[TABLE]

[TABLE]

where $v=dx^{1}\wedge dx^{2}\wedge dx^{3}\wedge dx^{4}$ is the volume form.

Using Proposition 2.5 , we have

Proposition 2.6

Let $U$ be an open subset of $\mathcal{H}$. If $f:U\to\mathcal{H}$ is a function given by

[TABLE]

where $x=\displaystyle\sum_{i=1}^{4}x_{i}e_{i}\in U$ with $x_{1}$, $x_{2}$, $x_{3}$, $x_{4}\in\mathcal{R}$, $f_{1}$ and $f_{0}$ are $\bf{C}^{1}$ functions, then the following are equivalent:

(i)

$f$* is algebraic regular on $U$;*

(ii)

Both the equation

[TABLE]

and the the equation

[TABLE]

hold on $U$.

3 Characterizing Algebraic Regularity by Difference Quotients

Let $f(x)=f_{1}(x_{1},x_{2},x_{3},x_{4})e_{1}+\displaystyle\sum_{k=2}^{4}x_{k}f_{0}(x_{1},x_{2},x_{3},x_{4})e_{k}$ be a quaternion-valued function defined on an open subset $U$ of $\mathcal{H}$, where $x=\displaystyle\sum_{i=1}^{4}x_{i}e_{i}$ with $x_{1}$, $x_{2}$, $x_{3}$, $x_{4}\in\mathcal{R}$ for $1\leq i\leq 4$. For each $c=\displaystyle\sum_{i=1}^{4}c_{i}e_{i}$ with $c_{1}$, $c_{2}$, $c_{3}$, $c_{4}\in\mathcal{R}$, we define six pure quaternion-valued functions on $U$ as follows:

[TABLE]

[TABLE]

where $2\leq i<j\leq 4$.

In the proposition below, we characterize the algebraic regularity by the limits of a new kind of difference quotients which use the six pure quaternion-valued functions $\stackrel{{\scriptstyle\leftarrow}}{{f}}^{c}_{i}(x)$ and $\stackrel{{\scriptstyle\rightarrow}}{{f}}^{c}_{i}(x)$ with $i=2$, $3$ and $4$.

Proposition 3.1

Let $f:U\to\mathcal{H}$ be a quaternion-valued function defined by

[TABLE]

where $U$ is an open subset of $\mathcal{H}$, $x=\displaystyle\sum_{k=1}^{4}x_{k}e_{k}\in U$ with $x_{1}$, $x_{2}$, $x_{3}$, $x_{4}\in\mathcal{R}$. Let $c=\displaystyle\sum_{k=1}^{4}c_{k}e_{k}\in U$ and $\Delta q=\displaystyle\sum_{k=1}^{4}(\Delta q)_{k}e_{k}$, where $c_{k}$, $(\Delta q)_{k}\in\mathcal{R}$ for $1\leq k\leq 4$. If $f_{1}(x_{1},x_{2},x_{3},x_{4})$ and $f_{0}(x_{1},x_{2},x_{3},x_{4})$ are $\mathcal{C}^{1}$ functions, then the following are equivalent:

(i)

$f$* is algebraic regular at $x=c$;*

(ii)

\displaystyle\lim_{\Delta q\to 0}(\Delta q)^{-1}\Big{\{}f(c+\Delta q)-f(c)+\displaystyle\sum_{i=2}^{4}\Big{[}\stackrel{{\scriptstyle\leftarrow}}{{f}}^{c}_{i}\Big{(}c+(\Delta q)_{i}\displaystyle\sum_{k=1}^{4}e_{k}\Big{)}-\stackrel{{\scriptstyle\leftarrow}}{{f}}^{c}_{i}(c)\Big{]}\Big{\}}* exists;*

(iii)

\displaystyle\lim_{\Delta q\to 0}\Big{\{}f(c+\Delta q)-f(c)+\displaystyle\sum_{i=2}^{4}\Big{[}\stackrel{{\scriptstyle\rightarrow}}{{f}}^{c}_{i}\Big{(}c+(\Delta q)_{i}\displaystyle\sum_{k=1}^{4}e_{k}\Big{)}-\stackrel{{\scriptstyle\rightarrow}}{{f}}^{c}_{i}(c)\Big{]}\Big{\}}(\Delta q)^{-1}* exists.*

Moreover, if one of the three coditions above holds, then both the limit in (ii) and the limit in (iii) equal to $\displaystyle\frac{\partial f}{\partial x_{1}}(c)$.

Based on Proposition 3.1, we call $\displaystyle\frac{\partial f}{\partial x_{1}}$ the quaternion derivative of an algebraic regular function $f$ on an open subset $U$ of $\mathcal{H}$. It is easy to check that if $f$ is an algebraic regular function on an open subset $U$ of $\mathcal{H}$, then its quaternion derivative $\displaystyle\frac{\partial f}{\partial x_{1}}$ is also an algebraic regular function on the open subset $U$.