Graffitti from 1843 Key to Mysteries Investigated in LHC

This "Supercomplex algebra" is called the Quaternion algebra. They are found in the algebra of three dimensional rotations (so they are, in a sense, at least, found in Quantum Mechanics in all cases involving spin [spin(3), so(3), and su(2)]). This is one of the simplest non-commutative algebras.

In fact, there are computer graphics systems that use quaternions.

The Standard Model uses an even larger algebra (u(1) + su(2) + su(3), where '+' is being used in place of the kronecker sum).

It sounds like Alain Connes has an approach using a fiber bundle space (which extends the tangent spaces of General Relativity) having a structure of a product of two Quaternion spaces (sort of like q + q, where 'q' is the appropriate Quaternion algebra). (Actually, since the new space extends the tangent space of the spacetime manifold, I suppose it would be something like q + q + R4.)

(One can go even farther, as a "Supercomplex algebra", to the Octonion algebra, which also looses associativity, in addition to commutativity. This is intimately related to the exceptional Lie algebras, which some consider to be the most beautiful mathematical structures known. In a sense, it would be a "shame" if the Universe were to settle for any lesser algebra than the largest [finite] exceptional Lie algebra: E₈.)

I would be very interested in what "good part of the zoo of particles found in particle accelerators" "exactly match" in this approach taken by Connes and his collaborators.

I kind of like at least some of the ideas of "noncommutative geometry". However, I certainly wouldn't start using the term "quantum geometry" until I got to the point of endorsing it as "the" basis upon which the Theory of Everything (ToE) "should" or "will" be built.

Just some of my thoughts on this matter.

David