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In the complex quaternions algebra $\mathbb{C}\otimes\mathbb{H}$, there're 8 elements: 1, $i$, $\hat i$, $\hat j$, $\hat k$ (quateronic ijk), $i\hat i$, $i\hat j$, $i\hat k$. The last three objects are claimed to generate rotations, and we could identify them as $\sigma_x$, $\sigma_y$, and $\sigma_z$, respectively. I'm wondering why this is the case? Is $i\hat i$ essentially a matrix? How can I understand them? Thanks:)

ZR-
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    Do $i\hat i$, $i\hat j$, and $i\hat k$ obey the same algebra as $\sigma_x$, $\sigma_y$, and $\sigma_z$? – G. Smith Jan 11 '21 at 04:27
  • @G. Smith Thanks for the comment! There's a commutation relation: $\left[\frac{i\hat i}{2},\frac{i\hat j}{2}\right] = i\frac{i\hat k}{2}$, I think this is analogous to $\left[\sigma_a,\sigma_b\right] = 2i\sigma_c$, but I'm not pretty sure why $i\hat i$ is a matrix, and how to understand the physical meaning in terms of quaternions. – ZR- Jan 11 '21 at 04:47
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    @Zhengrong the point is they don't need to be matrices. Their defining characteristic is the algebra that they obey. The space generated by the standard 2x2 Pauli matrices is just one representation of the $\mathfrak{su}(2)$ Lie algebra. – Nihar Karve Jan 11 '21 at 04:49
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    Related : Why is there this relationship between quaternions and Pauli matrices. In my answer look how we could define the basic quartenions $\mathbf{i}\boldsymbol{=-}i\sigma_1,\mathbf{j}\boldsymbol{=-}i\sigma_2,\mathbf{k}\boldsymbol{=-}i\sigma_3$ in equations (19). From these definitions it's reasonable to identify $i\mathbf{i}\boldsymbol{=}\sigma_1,i\mathbf{j}\boldsymbol{=}\sigma_2,i\mathbf{k}\boldsymbol{=}\sigma_3$. – Frobenius Jan 19 '21 at 14:23

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For now, just forget about how the names $i\hat{i}$, $j\hat{j}$ and $k\hat{k}$ arise - call them $a_1$, $a_2$ and $a_3$. As you have worked out for yourself, these obey the $\mathfrak{su}(2)$ Lie algebra$^\dagger$: $[\frac12 a_i, \frac12 a_j] = \frac12\epsilon_{ijk}a_k$ - note that we did not have to make reference to any matrices here, we simply used the algebraic relations from the quaternion algebra (in fact the Lie group of unit quaternions $\text{Sp}(1)$ is isomorphic to the Lie group $\text{SU}(2)$). You could then formally exponentiate these $a_i$ to get from the $\mathfrak{su}(2)$ Lie algebra to the $\text{SU}(2)$ group.

However as they stand, the $a_i$ are currently abstract and mathematical: the group alone only encodes how the actions or transformations relate to each other, but not how to apply them. So the next step is to select some objects that we would like to work with, and create "operators" (for lack of a better word) that preserve the information encoded by the group, but can act on our objects in a concrete manner - this is, as you will have guessed, a group representation (or analogously, a Lie algebra representation that preserves the information encoded at the tangent space of a Lie group). Of course, how these transformations work specifically will depend on the type of object under consideration.

For example, two-component complex spinors are often used in non-relativistic QM, so if we take our "objects" in this case to be two-dimensional complex vectors, a natural candidate for our "operators" is $2\times 2$ complex matrices (alternatively, working in the Lie algebra perspective, we also want $2\times 2$ complex matrices, since the exponentiation preserves matrix dimension). As it turns out, the Pauli matrices $\sigma_i$ are exactly compatible with this description, and obey the $\mathfrak{su}(2)$ commutation relations ("preserve the information encoded the group"), similarly, you could find representations that operate on higher-dimensional vectors, or even wackier algebraic objects.

So in their raw form, you cannot say that $i\hat{i}$, $j\hat{j}$ and $k\hat{k}$ "generate rotations" - what you must do is identify the Lie algebra, choose objects that you want to act on, form a concrete representation and exponentiate combinations of these representations to get concrete actions that can operate on the chosen objects. The fact that these representations can be used to effect rotations (among other things) comes from the association between the groups $\text{SU}(2)$ and $\text{SO}(3)$.

$^\dagger$ These commutation relations agree with those obeyed by the Pauli matrices up to a factor of $i$, since physicists prefer to work with self-adjoint operators.

Nihar Karve
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