Chapter 9: Groups over Other $\KK$

A Quaternionic Description of $SO(3)$

We saw in §7.2 how to generate $\SO(2)$ through multiplication by a unit-normed complex number. We can try the same thing with the quaternions: Multiplication by a unit-normed quaternion should be a rotation. Since the quaternions are four-dimensional, we expect to get rotations in four dimensions.

Not so fast! We saw in §3.4 that quaternionic multiplication corresponds to rotations in two planes, making it difficult to count. Let's try to generate rotations in just one plane.

We start with the imaginary quaternions, which correspond to vectors in three dimensions. If we want to rotate only these three dimensions, we need transformations that leave the real part of a quaternion alone. Consider the map \begin{equation} q \longmapsto pqp^{-1} \end{equation} which is called conjugation of $q$ by $p$. 1) If $[p,q]=0$, that is, if $p$ and $q$ commute, then of course $pqp^{-1}=q$. In particular, conjugation preserves $1$, and hence all real numbers.

Conjugation automatically preserves the norm of $q$, since \begin{equation} |pqp^{-1}| = |p|\,|q|\,|p|^{-1} = |q| \end{equation} This argument shows that the norm of $p$ plays no role in the transformation, so we can assume $|p|=1$. So let $p=e^{i\alpha}$. Then $p^{-1}=\bar{p}$, and conjugation by $p$ becomes \begin{equation} q \longmapsto pq\bar{p} \end{equation}

What does conjugation by $p$ do to $q$? Since $\HH=\CC\oplus\CC j$, we can write \begin{equation} q = r_1 e^{i\theta} + r_2 e^{i\phi} j \label{Hdecomp} \end{equation} Remembering that $i$ and $j$ anticommute, we therefore have \begin{equation} pq\bar{p} = e^{i\alpha} \left(r_1 e^{i\theta} + r_2 e^{i\phi} j\right) e^{-i\alpha} = r_1 e^{i\theta} + r_2 e^{i(\phi+2\alpha)} j \end{equation} Thus, conjugation by $e^{i\alpha}$ leaves both the real and $i$ directions alone, but induces a rotation by $2\alpha$ in the $jk$ plane. Similarly, conjugation by $e^{j\alpha}$ and $e^{k\alpha}$ correspond to rotations by $2\alpha$ in the $ki$ and $ij$ planes, respectively. Since we can generate any rotation by combining rotations in the coordinate planes, we see that conjugation by unit-normed quaternions generates $\SO(3)$, the rotations in $\Im\HH$. 2) In three dimensions, we can actually do better: any rotation in three dimensions is in fact a rotation about a single axis, so we can dispense with the notion of “generators” in this case. In other words, \begin{equation} \SO(3) = \{ p\in\HH : |p|=1 \} \end{equation} or, in the sense of transformations, \begin{equation} \SO(3) = \{ q \longmapsto pq\bar{p} : p,q\in\HH, |p|=1 \} \end{equation}

A special case occurs when $p=i$, corresponding to a rotation by $\pi$ in the $jk$ plane, which takes every element of the $jk$ plane and multiplies it by $-1$. We call such a transformation a flip. Any imaginary unit $u$ can be used here, corresponding to a flip in the plane perpendicular to it.

What about one-sided multiplication by unit-normed quaternions? These transformations are in 1-to-1 correspondence with conjugation, so there aren't enough transformations to generate all of $\SO(4)$. In fact, this must be another version of $\SO(3)$ — not the same version, since $1$ is not left invariant. Let's take a closer look.

Starting again from (\ref{Hdecomp}), we have \begin{equation} e^{i\alpha} \left(r_1 e^{i\theta} + r_2 e^{i\phi} j\right) = r_1 e^{i(\theta+\alpha)} + r_2 e^{i(\phi+\alpha)} j \end{equation} so that left multiplication by $e^{i\alpha}$ corresponds to a rotation by $\alpha$ in both the $1i$ and $jk$ planes. What about right multiplication? Now we have \begin{equation} \left(r_1 e^{i\theta} + r_2 e^{i\phi} j\right) e^{i\alpha} = r_1 e^{i(\theta+\alpha)} + r_2 e^{i(\phi-\alpha)} j \end{equation} corresponding to a rotation by $+\alpha$ in the $1i$ plane, but a rotation by $-\alpha$ in the $jk$ plane.

We conclude that there are three different versions of $\SO(3)$ here, induced respectively by conjugation, left multiplication, and right multiplication by a unit quaternion.

1) Do not confuse the two different uses of the word “conjugation”!
2) As noted above, the restriction on the norm can be dropped, since it cancels out of the transformation.

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