User:KSmrq/Ruminations

Regarding User:Fropuff/Draft 6 (on the unit quaternion group):

I see you wish to mention SO(4). There are, of course, two interesting connections to explore. One is to use a complex structure, as in

• Atiyah, Bott, & Shapiro. "Clifford modules", Topology 3, Supplement 1, July 1964, pp. 3–38, ISSN 0040-9383, doi:10.1016/0040-9383(64)90003-5 (also search the web).

Take three anticommuting matrices that each square to the negative identity,

$X={\begin{bmatrix}0&0&0&1\\0&0&-1&0\\0&1&0&0\\-1&0&0&0\end{bmatrix}},\qquad Y={\begin{bmatrix}0&0&1&0\\0&0&0&1\\-1&0&0&0\\0&-1&0&0\end{bmatrix}},\qquad Z={\begin{bmatrix}0&-1&0&0\\1&0&0&0\\0&0&0&1\\0&0&-1&0\end{bmatrix}},$

which along with the identity matrix duplicate the quaternions 1, i, j, and k. Here quaternion conjugation is matrix transposition. If we step back and use

$J={\begin{bmatrix}0&-1\\1&0\end{bmatrix}}$

and the 2×2 identity matrix to duplicate the complex 1 and i, then simply by using 2×2 block partitioning we see connections of interest.

$X={\begin{bmatrix}0&-J\\-J&0\end{bmatrix}},\qquad Y={\begin{bmatrix}0&I\\-I&0\end{bmatrix}},\qquad Z={\begin{bmatrix}J&0\\0&-J\end{bmatrix}}$

Noting that X, Y, and Z are all special orthogonal matrices (as is J), we have revealed unit quaternions within SO(4), and revealed SU(2) as well.

The second connection is the unusual coincidence, that SO(4) is the direct product of SO(3) and S³, which we may exhibit as follows. Let Q be the 4×4 matrix for one unit quaternion, and let R be another. First we recreate R⁴ as a 4×4 matrix V = wI+xX+yY+zZ. Then we note that RVR^T leaves wI unchanged and rotates the rest, so we have revealed SO(3). Finally, QRVR^T is enough to handle all of SO(4). One last piece of magic lets us move that R^T to the left of V while replacing V with a column vector; define

${\overleftarrow {X}}={\begin{bmatrix}0&0&0&1\\0&0&1&0\\0&-1&0&0\\-1&0&0&0\end{bmatrix}},$

transposing the upper left 3×3 corner, and similarly for any 4×4 matrix. Then

$(Q,R)\mapsto (Q,R{\overleftarrow {R}}^{T})\mapsto QR{\overleftarrow {R}}^{T}\,\!$

gives explicit mappings from S³×S³ to S³×SO(3) to SO(4).

--KSmrq^T 08:29, 4 May 2007 (UTC)