Implications of $\text{Spin}(3)\times \text{Spin}(3) \cong \text{Spin}(4)$

Question

I recently learned that $\text{Spin}(3)\times \text{Spin}(3) \cong \text{Spin}(4)$, and find it intriguing: it seems like the sort of thing that would have lots of interesting consequences. What are the implications of this isomorphism? I'd be interested in connections between this and any other area of math (or physics!).

Answer 1

I'm guessing an answer that best discusses the "implications" of this accidental isomorphism from a physical perspective is one that discusses the implications in spin geometry specifically. But that area is deeper than my expertise so I unfortunately can't comment on it. Instead, I can mention a bunch of ideas and facts that, are not really implications per se but go hand-in-hand with this isomorphism.

First, though, there is one elementary implication, and that is in classifying simplie Lie groups and algebras. The characterization of Lie groups splits into covering space theory and Lie algebras, and the characterization of Lie algebras splits into solvable and semisimple ones, and the characterization of semisimple ones is simply a matter of classifying simple Lie algebras. (I am talking about complex ones, there is some more to talk about going between real and complex algebras/groups.) The classification of simple Lie algebras is known, and simple enough to cover in a semester or two of graduate class. The case of $\mathfrak{so}(4)$ is the only $\mathfrak{so}(n)$ that is not simple. There are other accidental isomorphisms as well between simple lie algebras.

Another fact is how quaternions are used to model 3D and 4D rotations. Conjugating vectors by versors yields rotations (and there is an angle-doubling effect which highlights the spinorial nature of this homomorphism), and left/right-regular multiplications by pairs of versors yields arbitrary 4D rotations. The former is useful in graphics and computational models of physical orientation. I am not sure how useful the latter is - even ignoring the gravitational curvature of general relativity, the spacetime of special relativity has three spatial and one temporal dimension (rather than four spatial dimensions), and in this context there is an accidental isomorphism between $\mathrm{SL}_2\mathbb{C}$ and $\mathrm{Spin}(3,1)$ (which extends the one between $\mathrm{SU}(2)$ and $\mathrm{Spin}(3)$).

This is also useful for describing left and right isoclinic rotations. All rotations in higher dimensions restrict to plane rotations in a bunch of orthogonal (2D) planes. (Basically, the real version of the spectral theorem.) The orientations of the plane rotations either "add up" to the orientation of the whole space, or the opposite orientation - this creates a distinction between left/right isoclinic rotations in even dimensions (there is no such distinction in odd dimensions since the planes can't "add up" to the whole space anyway). In 4D, the left/right isoclinic rotations come from left/right multiplication by versors. In higher dimensions, the left/right isoclinic rotations don't commute and don't form subgroups (although they are unions of one-parameter subgroups, so are images of the exponential map).

One way to continue the progression $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ is with octonions $\mathbb{O}$. These are Euclidean / Hurwitz / composition / normed division algebras (there are lots of names). But another way is with Clifford algebras. The Clifford algebra $\mathrm{C}\ell(p,q)$ is generated by $p$ and $q$ anticommuting square roots of $+1$ and $-1$ (respectively - although sign conventions differ). So $\mathbb{R},\mathbb{C},\mathbb{H}$ are $\mathrm{C}\ell(n,0)$ for $n=0,1,2$. We define the spin groups $\mathrm{Spin}(p,q)$ as generated by (products of evenly many) unit vectors in $\mathrm{C}\ell(p,q)$, these are double covers of $\mathrm{SO}(p,q)$ (mod $\pi_0$, with a couple exceptions). Spin representations come from modules over Clifford algebras. We can classify Clifford algebras recursively - by tensoring with $\mathrm{C}\ell(p,q)$ with $(p,q)=(2,0),(1,1),(0,2)$, which are $\mathbb{H}$, $\mathbb{R}(2)$, $\mathbb{R}(2)$ (the latter two are $2\times2$ matrix algebras). A microcosm of this is the $\otimes$ table

$$ \begin{array}{c|c|c|c} \otimes & \mathbb{R} & \mathbb{C} & \mathbb{H} \\ \hline \mathbb{R} & \mathbb{R} & \mathbb{C} & \mathbb{H} \\ \mathbb{C} & & \mathbb{C}^2 & \mathbb{C}(2) \\ \mathbb{H} & & & \mathbb{R}(4) \end{array} $$

Note $\mathbb{H}\otimes\mathbb{H}\cong\mathbb{R}(4)$ is a "linearization" of $S^3\times_{\mathbb{Z}_2} S^3\cong \mathrm{SO}(4)$. So you could say this isomorphism (linearized) has implications for the classification of Clifford algebras, hence construction of spin representations (and also subsequently, for constructing maximal trivial subbundles of sphere's tangent bundles, IOW linearly independent vector fields on spheres).

The (real, oriented) Grassmanian $\mathrm{Gr}(n,k)$ is the space of all (oriented) $k$-dimensional subspaces of $\mathbb{R}^n$. It can be considered a subspace of $\Lambda^k\mathbb{R}^n$. (The unoriented Grassmanian is a quotient by $\mathbb{Z}_2$, and may be considered a subspace of $\mathbb{P}(\Lambda^k\mathbb{R}^n)$.) The first nontrivial Grassmanian would be $\mathrm{Gr}(4,2)$. In the case of $k=2$, we can identify $\Lambda^2\mathbb{R}^n\cong\mathfrak{so}(n)$ (as representations). The fact $\mathrm{so}(4)\cong\mathfrak{so}(3)\times\mathfrak{so}(3)$ corresponds to a decomposition $\Lambda^2=\Lambda^2_+\oplus\Lambda^2_-$. The Hodge star $\ast:\Lambda^k\mathbb{R}^n\to\Lambda^{n-k}\mathbb{R}^n$ (which linearly extends the orthogonal complement function $\mathrm{Gr}(n,k)\to\mathrm{Gr}(n,n-k)$) acts as a linear operator on $\Lambda^2\mathbb{R}^4$, whose $\pm1$ eigenspaces are $\Lambda^2_\pm$. The decomposition $\mathrm{Gr}(4,2)=S^2\times S^2$ exists within $\Lambda^2=\Lambda^2_+\oplus\Lambda^2_-$. All of this left/right-handedness and pairedness goes hand-in-hand with $\mathrm{Spin}(4)\cong\mathrm{Spin}(3)\times\mathrm{Spin}(3)$.