Riemann Sphere vs. Quaternions - How are they different, and similar?

Question

Simply as a thought exercise, I'm curious in looking at how the Riemann Sphere and the Quaternions are different, and what they could possibly have in common?

Before explaining, yes, I understand that these objects are of vastly different natures, seemingly unrelated. But, math does seem to be a little bit about sometimes going where things don't seem to make sense. Maybe it'd just be fun to discuss?

Motivation - The Riemann Sphere (RS) and Quaternions are both a 'step up' from the Complex Plane in a certain sense. Putting complex functions on the Riemann sphere does seem to offer an entirely new perspective. (Unit disk in Complex Plane becomes Southern Hemisphere of RS, and outside of unit disk becomes Northern Hemisphere. This, in a sense, shows that the unit disk is as 'large' as the rest of the Complex Plane, but in a way that becomes clearly obvious)

On the other hand, the Complex Plane forms a substructure of the Quaternions, so there is already an intimate relationship between them in this case as well. We could even potentially say that the Quaternions are stitched together from different Complex units, all with the same property as our usual i.

One could argue that the RS is essentially the Complex Plane, with little to no additional structure. However, the one-point compactification used to produce the RS from the Complex Plane does allow new possibilities in the RS, such as the ability to define 1/0.

I hope that the audience will forgive the hazy language, as I am not a professional mathematician. However, maybe this is enough to at least get a discussion going?

Someone also please tell me if this question is just utter nonsense, or if it belongs in a different category.

Answer 1

This is an extremely vague question; however, fortunately, there is at least one relatively precise thing that one can say here. Namely, the Riemann sphere $\mathbb{CP}^1$ is naturally acted on by the special unitary group $SU(2)$, which is the double cover of the group of orientation-preserving isometries of the Fubini-Study metric, and $SU(2)$ also both naturally occurs as a subgroup of and naturally acts on the quaternions $\mathbb{H}$: in fact it is isomorphic to the group of unit quaternions $Sp(1)$. This is an example of an exceptional isomorphism between two Lie groups.

Since $\mathbb{CP}^1$ with the Fubini-Study metric is just the $2$-sphere $S^2$ with the usual metric inherited from $\mathbb{R}^3$ (up to scale), its group of orientation-preserving isometries can also be identified with the special orthogonal group $SO(3)$; this exhibits another exceptional isomorphism, where $SU(2) \cong Sp(1)$ is also isomorphic to a group called the Spin group $Spin(3)$ which is the double cover of $SO(3)$. This is part of a long story involving spinors and Clifford algebras and other fun stuff like that.