A Most Perplexing Algebra
Originally, I had been enthralled by how Macfarlane’s quaternions effortlessly yielded a Minkowskian quadratic form and Lorentz group generators via left/right multiplication by \(i, j, k\). Yet, I was baffled by the failure for those six matrices to close under multiplication. After writing a snippet of code, I determined that they closed under repeated multiplication to a 64-element closure.
Non-Associative \(\rightarrow\) No Global Basis
I noticed that my choice of basis assignment \(\lbrace e_0, e_1, e_2, e_3 \rbrace\) such as \(\lbrace 1, i, j, k \rbrace \) or \(\lbrace k, j, i, 1 \rbrace\) led to a different six matrix representations but that they led inevitably back to the same 64. I also noticed that what in one basis was the representative of a single multiplication might be a composite operation in another.
What I concluded based off of this was that unlike Hamilton’s quaternions, it was not possible to establish a single choice of basis and have it remain closed under repeated multiplicative operations. After all, I was working with a non-associative algebra (Macfarlane’s), and matrix multiplication itself is associative. Thus, \(M_a M_b \ne M_{ab}\). I would get additional actions in that closure which were the composite multiplications where order of application mattered.
A Groupoid to Map Between Them
Because of this, I began considering the use of groupoids where the objects were basis assignments and the arrows were the allowed transformations (given by the six matrix representations of left/right multiplication). This seemed all well and good when I lived in a land of pure permutations.
Unleash the Projectors
Alas, to globally glue local choices of basis representations, transitions must occur. This is no troublesome affair when shifting between \(i\) or \(j\) or \(k\), but throw \(1\) into the ring, and trouble ensues. What’s the trouble with identity, you might ask?
Idempotents.
\[\frac{(1+\hat{u})}{2}\]
Lovely little blighters they are. For it isn’t merely that they multiply to themselves, no no… They act as projectors, splitting the space in twain, reducing the independent degrees of freedom and sending bits and bobs off to the kernel.
In Pursuit of Shifting Contexts
So no, it wasn’t enough to consider groupoids what with their invertible arrows and all. They would never do to represent an algebra with non-invertible transformations. It was time to trade inverses for adjunctions and equals for equivalences.
Now the delightful hunt begins for the appropriate mathematical formalism. Pip pip and tally ho!
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