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## Motivation.

Suppose that we want to represent GA(2,0) (Euclidean) multivectors as a pair of complex numbers, with a structure like

\begin{equation}\label{eqn:bicomplexGA20:20}

M = (m_1, m_2),

\end{equation}

where

\begin{equation}\label{eqn:bicomplexGA20:40}

\begin{aligned}

\gpgrade{M}{0,2} &\sim m_1 \\

\gpgrade{M}{1} &\sim m_2.

\end{aligned}

\end{equation}

Specifically

\begin{equation}\label{eqn:bicomplexGA20:60}

\begin{aligned}

\gpgrade{M}{0} &= \textrm{Re}(m_1) \\

\gpgrade{M}{1} \cdot \Be_1 &= \textrm{Re}(m_2) \\

\gpgrade{M}{1} \cdot \Be_2 &= \textrm{Im}(m_2) \\

\gpgrade{M}{2} i^{-1} &= \textrm{Im}(m_1),

\end{aligned}

\end{equation}

where \( i \sim \Be_1 \Be_2 \).

## Multivector product representation.

Let’s figure out how we can represent the various GA products, starting with the geometric product. Let

\begin{equation}\label{eqn:bicomplexGA20:80}

\begin{aligned}

M &= \gpgrade{M}{0,2} + \gpgrade{M}{1} = (m_1, m_2) = (m_{11} + m_{12} i, m_{21} + m_{22} i) \\

N &= \gpgrade{N}{0,2} + \gpgrade{N}{1} = (n_1, n_2) = (n_{11} + n_{12} i, n_{21} + n_{22} i),

\end{aligned}

\end{equation}

so

\begin{equation}\label{eqn:bicomplexGA20:200}

\begin{aligned}

M N

&= \gpgrade{M}{0,2} \gpgrade{N}{0,2} + \gpgrade{M}{1} \gpgrade{N}{1} \\

&\quad+ \gpgrade{M}{1} \gpgrade{N}{0,2} + \gpgrade{M}{0,2} \gpgrade{N}{1}

\end{aligned}

\end{equation}

The first two terms have only even grades, and the second two terms are vectors. The complete representation of the even grade components of this multivector product is

\begin{equation}\label{eqn:bicomplexGA20:240}

\gpgrade{M N}{0,2} \sim \lr{ m_1 n_1 + \textrm{Re}(m_2 n_2^\conj) – i \textrm{Im}(m_2 n_2^\conj), 0 },

\end{equation}

or

\begin{equation}\label{eqn:bicomplexGA20:260}

\begin{aligned}

\gpgrade{M N}{0} &= \textrm{Re}\lr{ m_1 n_1 + m_2 n_2^\conj } \\

\gpgrade{M N}{2} i^{-1} &= \textrm{Im}\lr{ m_1 n_1 – m_2 n_2^\conj }.

\end{aligned}

\end{equation}

For the vector components we have

\begin{equation}\label{eqn:bicomplexGA20:280}

\begin{aligned}

\gpgrade{M N}{1}

&=

\gpgrade{M}{1} \gpgrade{N}{0} + \gpgrade{M}{0} \gpgrade{N}{1}

+

\gpgrade{M}{1} \gpgrade{N}{2} + \gpgrade{M}{2} \gpgrade{N}{1} \\

&= \gpgrade{M}{1} n_{11} + m_{11} \gpgrade{N}{1} + \gpgrade{M}{1} i n_{12} + i m_{12} \gpgrade{N}{1}.

\end{aligned}

\end{equation}

For these,

\begin{equation}\label{eqn:bicomplexGA20:300}

\begin{aligned}

\gpgrade{M}{1} i

&= \lr{ m_{21} \Be_1 + m_{22} \Be_2 } \Be_{12}

&= – m_{22} \Be_1 + m_{21} \Be_2,

\end{aligned}

\end{equation}

and

\begin{equation}\label{eqn:bicomplexGA20:320}

\begin{aligned}

i \gpgrade{N}{1}

&= \Be_{12} \lr{ n_{21} \Be_1 + n_{22} \Be_2 }

&=

n_{22} \Be_1 – n_{21} \Be_2.

\end{aligned}

\end{equation}

Comparing to

\begin{equation}\label{eqn:bicomplexGA20:340}

i (a + i b)

= -b + i a,

\end{equation}

we see that

\begin{equation}\label{eqn:bicomplexGA20:360}

\gpgrade{M N}{1}

\sim

\lr{ 0, n_{11} m_2 + m_{11} n_2 + n_{12} i m_2 – m_{12} i n_2 }.

\end{equation}

If we want the vector coordinates, those are

\begin{equation}\label{eqn:bicomplexGA20:380}

\begin{aligned}

\gpgrade{M N}{1} \cdot \Be_1 &= \textrm{Re} \lr{ n_{11} m_2 + m_{11} n_2 + n_{12} i m_2 – m_{12} i n_2 } \\

\gpgrade{M N}{1} \cdot \Be_2 &= \textrm{Im} \lr{ n_{11} m_2 + m_{11} n_2 + n_{12} i m_2 – m_{12} i n_2 }.

\end{aligned}

\end{equation}

## Summary.

\begin{equation}\label{eqn:bicomplexGA20:n}

M N \sim

\lr{ m_1 n_1 + \textrm{Re}(m_2 n_2^\conj) – i \textrm{Im}(m_2 n_2^\conj), n_{11} m_2 + m_{11} n_2 + n_{12} i m_2 – m_{12} i n_2 }.

\end{equation}

A sample Mathematica implementation is available, as well as an example notebook (that also doubles as a test case.)

## Clarification.

I skipped a step above, showing the correspondances to the dot and wedge product.

Let \(z = a + bi\), and \(w = c + di\). Then:

\begin{equation}\label{eqn:bicomplexGA20:420}

\begin{aligned}

z w^\conj

&=

\lr{ a + b i } \lr{ c – d i } \\

&= a c + b d -i \lr{ a d – b c }.

\end{aligned}

\end{equation}

Compare that to the geometric product of two vectors \( \Bx = a \Be_1 + b \Be_2 \), and \( \By = c \Be_1 + d \Be_2 \), where we have

\begin{equation}\label{eqn:bicomplexGA20:440}

\begin{aligned}

\Bx \By &= \Bx \cdot \By + \Bx \wedge \By \\

&= \lr{ a \Be_1 + b \Be_2 } \lr{ c \Be_1 + d \Be_2 } \\

&= a c + b d + \Be_1 \Be_2 \lr{ a d – b c }.

\end{aligned}

\end{equation}

So we have

\begin{equation}\label{eqn:bicomplexGA20:460}

\begin{aligned}

a b + cd &= \Bx \cdot \By = \textrm{Re} \lr{ z w^\conj } \\

a d – b c &= \lr{ \Bx \wedge \By } \Be_{12}^{-1} = – \textrm{Im} \lr{ z w^\conj }.

\end{aligned}

\end{equation}

We see that \( \lr{z w^\conj}^\conj = z^\conj w \) can be used as a representation of the geometric product of two vectors (setting \( i = \Be_1 \Be_2 \) as usual.)

## Another simplification.

We have sums of the form

\begin{equation}\label{eqn:bicomplexGA20:480}

\textrm{Re}(z) w \pm \textrm{Im}(z) i w

\end{equation}

above. Let’s see if those can be simplified. For the positive case we have

\begin{equation}\label{eqn:bicomplexGA20:500}

\begin{aligned}

\textrm{Re}(z) w + \textrm{Im}(z) i w

&=

\inv{2} \lr{ z + z^\conj } w + \inv{2} \lr{ z – z^\conj } w \\

&=

z w,

\end{aligned}

\end{equation}

and for the negative case, we have

\begin{equation}\label{eqn:bicomplexGA20:520}

\begin{aligned}

\textrm{Re}(z) w – \textrm{Im}(z) i w

&=

\inv{2} \lr{ z + z^\conj } w – \inv{2} \lr{ z – z^\conj } w \\

&=

z^\conj w.

\end{aligned}

\end{equation}

This, with the vector-vector product simplification above, means that we can represent the full multivector product in this representation as just

\begin{equation}\label{eqn:bicomplexGA20:540}

M N \sim

\lr{ m_1 n_1 + m_2^\conj n_2, m_2 n_1 + m_1^\conj n_2 }.

\end{equation}