imaginary scator algebra

Submitted by mfg on Fri, 03/11/2017 - 14:18

The imaginary or elliptic scator set is represented with a minus subscript in the following way, $\mathbb{S}_{-}^{1+n}$. These numbers have an algebra of their own. A relevant feature of this algebra is that the product is not distributive over addition. In the additive representation a scator is written as

\begin{equation}
\overset{o}\varphi=f_{0} + \sum_{j=1}^{n}{f_{j}\check{\mathbf{e}}_{j} } ,
\end{equation}with $f_{0}, f_{j} \in \mathbb{R}$ for all $j$ from 1 to $n$ and  $\check{\mathbf{e}}_{j} \notin \mathbb{R}$. Scator elements have a scalar component and $n$ director components.

imaginary scator sum

Let two scators be $\overset{o}\alpha=a_{0} + \sum_{j=1}^{n}{a_{j}\check{\mathbf{e}}_{j}}$ and $\overset{o}\beta=b_{0} + \sum_{j=1}^{n}{b_{j}\check{\mathbf{e}}_{j}}$. Their sum is performed component by component,
\begin{equation}
\overset{o}\gamma = g_{0} + \sum_{j=1}^{n}{g_{j}\check{\mathbf{e}}_{j}} = \overset{o}\alpha + \overset{o}\beta = (a_{0} + b_{0}) + \sum_{j=1}^{n}({a_{j}+ b_{j})\check{\mathbf{e}}_{j}}.
\end{equation}We can see that the scalar part of the sum is the sum of the scalar components of scators  $\overset{o}\alpha$ y $\overset{o}\beta$. In a similar fashion, the director part of the sum is equal to the sum of the director coefficients with equal unit director component $\check{\mathbf{e}}_{j}$.

imaginary scator product

The product of two scators can be divided in three cases:

Case 1

Let $\overset{o}\alpha$ y $\overset{o}\beta$ be two scators such that $a_{0}b_{0}\neq 0$. Their product $\overset{o}\gamma=\overset{o}\alpha\overset{o}\beta$ is then defined by \begin{equation}\overset { o }\gamma   ={ a }_{ 0 }{ b }_{ 0 }\prod _{ k=1 }^{ n }{ \left( 1-\frac { { a }_{ k }{ b }_{ k } }{ { a }_{ 0 }{ b }_{ 0 } }  \right) +{ a }_{ 0 }{ b }_{ 0 }\sum _{ j=1 }^{ n }{ \left[ \prod _{ k\neq j }^{ n }{ \left( 1-\frac { { a }_{ k }{ b }_{ k } }{ { a }_{ 0 }{ b }_{ 0 } }  \right) \left( \frac { { a }_{ j } }{ { a }_{ 0 } } +\frac { { b }_{ j } }{ { b }_{ 0 } }  \right)  }  \right]  }  } { \check {\mathbf{ e }  }_{ j }}.\end{equation}

Case 2

If one of the factors has only one, possibly null, director component and zero scalar component $a_{0}=0, b_{0}\neq 0$, say $\overset{o}\alpha={a}_{l} { \check {\mathbf{ e } }_{ l }}$ and the other scator factor is $\overset{o}\beta=b_{0} + \sum_{j=1}^{n}{b_{j}}{\check{\mathbf{e}}_{j}}$, such that $b_{0}\neq 0$, their product is then defined as\begin{equation}\overset { o }{ \gamma  } =\left(- { a }_{ l }{ b }_{ l } \right) +\left( { a }_{ l }{ b }_{ 0 } \right) { \check { \mathbf{ e }  }_{ l }}-\sum _{ j\neq l }^{ n }{ \left( \frac { { a }_{ l }{ b }_{ l }{ b }_{ j } }{ { b }_{ 0 } }  \right)  } { \check {\mathbf{ e }  }_{ j }}.\end{equation}

Case 3

Finally, if both factors have zero additive scalar component, $a_{0}=0, b_{0}= 0$ that is,  $\overset{o}\alpha={a}_{l} { \check {\mathbf{ e } }_{ l }}$ and $\overset{o}\beta={b_{j}\check{\mathbf{e}}_{j}}$. Their product is then\begin{equation}\overset { o }{ \gamma  } =({a}_{l}  { \check {\mathbf{  e }  }_{ l }})({b_{j}\check{\mathbf{e}}_{j}})=-{a}_{l} b_{j} \delta_{lj},\end{equation}

It is possible to obtain the following case from the previous ones:

Let two scators have an arbitrary scalar part and at most one director component different from zero, $\overset{o}\alpha={a}_{0}+{a}_{l} { \check {\mathbf{ e } }_{ l }}$ and $\overset{o}\beta={b}_{0}+{b}_{j} { \check {\mathbf{e} }_{ j }}$. Their product is then\begin{equation}\overset { o }{ \gamma  } ={ a }_{ 0 }{ b }_{ 0 }-{ a }_{ l }{ b }_{ j }{ \delta  }_{ lj }+{ b }_{ 0 }{ a }_{ l }{ \check { \mathbf{ e }  }_{ l }}+{ a }_{ 0 }{ b }_{ j }{ \check {\mathbf{  e }  }_{ j }}.\end{equation} If $l=j$, this result reproduces the product of complex numbers.

If the scator $\overset{o}\alpha$ has only a non vanishing scalar component and all its director components are zero, $\overset{o}\alpha={a}_{0}$, from case 1,

\begin{equation}{a}_{0}\overset{o}\beta={a}_{0}b_{0} + \sum_{j=1}^{n}{a}_{0}{b_{j}\check{\mathbf{e}}_{j}}\end{equation}

then, $\overset{o}\alpha={a}_{0}$ is a scaling factor for the scator $\overset{o}\beta$. Thus, the first component of a scator is labeled the scalar component.