Unit quaternions as rotations

Question

How would one represent the map $f$ such that $f(1) = i, f(i) = -1$ and keeping $j$ and $k$ fixed as a quaternion representation of rotations?

Answer 1

Well, there is a small question of what rotation representation you're talking about, but I think I've covered all the bases.

The only representation that is going to be relevant is the representations rotations of $\Bbb R^4$ represented as pairs of unit quaternions.

By solving the system of equations $q1s=i$, $qis=-1$, $qjs=j$ and $qks=k$, one arrives at the solution $q=\cos(\pi/4)+i\sin(\pi/4)$ and $s=\sin(\pi/4)+i\cos(\pi/4)$. Then the mapping $T(x)=qxs$ from $\Bbb H\to \Bbb H$ is modeling that particular rotation on $\Bbb R^4$.

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If you're talking about letting a quaternions act as a rotation on $\Bbb R^4 $ via left multiplication (the left regular representation), then you are looking for the matrix $\begin{bmatrix}0&-1&0&0\\1&0&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}$, but that is not in the regular representation of $\Bbb H$.

If instead you are thinking of the usual representation of rotations of 3-space, one can't do such a thing: conjugating by unit quaternions maps the reals into the reals, so it can't map 1 to i.