This playful notebook entry is a story that came to mind after a very real algebraic surprise I ran into while working with matrix representations.
Posh Properties
There once was an algebra that loved to try on matrix representations. It seemed of a reasonable and regal character, governing by simple laws and properties. While the algebra cherished its elements, it had but a scant four of them: \(\lbrace 1, i, j, k \rbrace \) which behaved in a most sensible fashion:\[i^2 = j^2 = k^2 = 1, \\ ij = -ji = k \\ jk=-kj = i, \\ ki = -ik = j. \]
The only fault of this algebra was that it was non-associative and thus its subjects had to be careful of the order in which they placed requests upon it. One day, the algebra’s advisor suggested that putting on a simple linear representation might help communicate the algebra’s laws and make it easier for everyone to follow them without getting confused. The algebra approved of this idea and decreed that the appropriate matrix representations be fashioned right away.
Layers of Finery
At first, the algebra commanded, “Bring me a matrix representation for i.” The tailors attempted to cut one of the proper shape but found that the pattern was chiral and could only be applied on from the left or the right. The algebra sighed but submitted to trying on a matrix representation for i that slid on from the left and another that slid on from the right.
Why, either way, this pattern looked quite smart and the algebra soon realized a clever trick: If it slid on an i from the left and then an i from the right, the matrix representation became identity and returned the algebra to its usual stately appearance.
The same trick worked with both j and k, and the algebra was terribly pleased at its new matrix representations. The algebra hung up all six in its wardrobe and beheld each chiral pair.
\[M_i^L = \begin{pmatrix} 0&1&0&0\\1&0&0&0\\0&0&0&-1\\0&0&1&0\end{pmatrix} \quad M_i^R = \begin{pmatrix} 0&1&0&0\\1&0&0&0\\0&0&0&1\\0&0&-1&0\end{pmatrix}\]
\[M_j^L = \begin{pmatrix} 0&0&1&0\\0&0&0&1\\1&0&0&0\\0&-1&0&0\end{pmatrix} \quad M_j^R = \begin{pmatrix} 0&0&1&0\\0&0&0&-1\\1&0&0&0\\0&1&0&0\end{pmatrix}\]
\[M_k^L = \begin{pmatrix} 0&0&0&1\\0&0&-1&0\\0&1&0&0\\1&0&0&0\end{pmatrix} \quad M_k^R = \begin{pmatrix} 0&0&0&1\\0&0&1&0\\0&-1&0&0\\1&0&0&0\end{pmatrix}\]
Then, the algebra closed the doors on its new representations and retired to its royal chambers for a good long sleep.
A Dream of Operadic Joy
In the depths of slumber, our algebra found itself standing before the wardrobe, doors open, all six matrix representations on display. Attendants applied i from the left and j from the right. And a new matrix representation shown forth from the layering! This pleased the algebra who immediately asked to try them on but switch the order of application. This resulted in a different composite, which led to further delight!
And then the algebra, knowing itself well, realized that not only did it not commute, it was also non-associative. And thus, ordering of representations should matter. Why, it could think of an infinite way to order the application of matrix representations! It would never run out of beautiful ways to display its decrees or intentions in only the finest of fashions.
Of course, all good things must come to an end, and the algebra soon awoke from that wonderful dream the next day with a mind set racing with all the beautiful compositions it might concoct.
Fashion Frustration
Before even partaking of breakfast, the algebra summoned the tailors and had them craft copies of the six matrix representations and began applying layer after layer in varying orders. At first, new combination after new combination came forth to the algebra’s great pleasure! However, it soon became apparent that new combinations became rarer and rarer and eventually no longer arose despite the tailors’ most fevered attempts at trying various orderings of a great many layers of matrices.
At first perturbed, then annoyed, then growing near a rage, the algebra commanded its tailors to stop messing about and find some way to craft a new matrix representation instead of repeating the old ones.
The tailors whispered among themselves that they’d exhaustively proved closure and that they knew that no more than 64 unique results existed from layering on matrix representations. It seemed there was nothing they could do to escape the algebra’s wrath.
That was until one tailor suddenly had a bright idea. “Let us stitch together a matrix representation that is half identity and half left i. This will yield something new for the algebra to try on and buy us some time to find a better solution.”
The other tailors agreed heartily and crafted \(\frac{1+i}{2}\) and also \(\frac{1-i}{2}\) for good measure.
The Grand Parade
The algebra was so very impressed by these two new forms, that it declared it would host a grand parade to show off each and every new fashionable form in every combination and end the whole affair by showing off the “half suits”.
Overall, the parade was a grand and glorious affair fit to go down in the memory books. The algebra prepared to take to the stage for a speech in the \(\frac{1+i}{2}\) representation. The crowd oohed and aahed over the clever split, and when the algebra promised to layer on a paired half suit, excitement grew fit to bursting.
In contrast, the clever tailor who came up with the suit was deeply troubled for he saw in his own handiwork a very unnerving sign. \[\frac{(I+M_i^L)}{2} = \frac{1}{2} \begin{pmatrix} 1&1&0&0\\1&1&0&0\\0&0&1&-1\\0&0&1&1\end{pmatrix}\]
The matrix had two matching rows and columns.
It was non-invertible.
How ever would they take the matrix representation off? Well, it was too late now. The crowd was chanting and cheering, and the algebra waved for the tailor to bring over the \(\frac{1-i}{2}\) representation and layer it on top.
The tailor gulped, carried the representation onstage, and slid it onto the algebra.
An Uninvertible Catastrophe
As soon as the second matrix representation slid over the first, the crowd audibly gasped in shock and horror. For the two had combined into nothing on top, leaving the algebra half bare before them all.
\[\frac{(I+M_i^L)}{2} \frac{(I-M_i^L)}{2} = \frac{1}{4} \begin{pmatrix} 1&1&0&0\\1&1&0&0\\0&0&1&-1\\0&0&1&1\end{pmatrix} \begin{pmatrix} 1&-1&0&0\\-1&1&0&0\\0&0&1&1\\0&0&-1&1\end{pmatrix}\] \[= \frac{1}{2} \begin{pmatrix} 0&0&0&0\\0&0&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}\]
And so, the algebra fled from the stage in embarrassment, and I’m sorry to report that it developed a tangent complex following the incident.
It all just goes to show a good lesson: When your own pride becomes too idempotent, be careful not to projector your insecurities to the audience.
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