### Cubes Galore

When multiplying $2 \times 2$ matrices, a pair of rows gives two multiplications and then an addition. By analogy, a cubic product on three operations would take a pair of faces, with the nonassociative product on these faces followed by the final addition. This gives which easily generalises to the $n \times n$ case. Alternatively, with one type of product a natural ternary product on three cubes $A$, $B$ and $C$ is given by

$(A,B,C)_{ijk} = \sum_{pqr} A_{ipq} B_{pjr} C_{qrk}$

$(A,B,C)_{ijk} = \sum_{pqr} A_{ipq} B_{pjr} C_{qrk}$

## 8 Comments:

Thanks very much for the reference to the paper

math-ph/0004031 by Richard Kerner in which he said:

"... the two copies of the Hilbert space that have been used to produce general linear operators ... L(V,V) ... (containing the algebra of observables) by means of the tensor product is utterly different from the role of the third copy ... V ... serving as the space of states ...".

This reminds me of what Jaak Lohmus, Eugene Paal, and Leo Sorgsepp said in their book "Nonassociative Algebras in Physics" (Hadronic Press 1994):

"... A general element ... of the binary sedenion algebra is ... represented by 16x16 matrices

...

We can introduces a mixed representation where one sedenion in the product is represented by a 16x16-matrix and the other one, by a 16-column.

for the ternary algebra this gives a nontrivial possibility to construct a ... mixed representation for ternary sedenions. In this representation two secenions A, C, ... are represented by ... 16x16-matrices and the third one, B, ... by 16-columns ...".

This seems physically interesting to me because

16x16-matrices can represent the Cl(8) Clifford algebra

and

16-columns can represent Cl(8) spinors.

Have you any ideas about use of sedenions as concrete examples of ternary algebraic structures?

Tony

PS - I see that Richard Kerner dedicated that paper to Andrzej Trautman, saying:

"... During the few years between 1964 and 1968, Andrzej Trautman taught me the modern and unifying approach to General Relativity and Gauge Theories, expressed by the new means of Differential Geometry: the theory of Fibre Bundles and Connections ... Fibre bundles were constructed as differential manifolds containing both the external and the internal spaces. The so-called external space was the observed four-dimensional space-time, while the internal space was supposed to carry the internal symmetries ...".

Andrzej Trautman did work with Meinhard Mayer (now emeritus at U.C. Irvine) showing how a Higgs mechanism emerges naturally from the geometry of dimensional reduction of spacetime (as in Kaluza-Klein models that I like), using results in the classic books by Kobayashi and Nomizu,

and it is nice to see a a paper dedicated to him.

Have not been thinking about sedenions, Tony. As you know, from my point of view, the relevant point about Clifford algebras is their universality, which is a general categorical property. Dim 16 just suggests to me a 4 qudit space.

About properties of Clifford algebras other than the general categorical property of universality,

for example,

the 8-periodicity of Real Clifford algebras and the 2-periodicity of Complex Clifford algebras,

do you regard them

as useful indicators of fundamental structure

or

as "Mostly Harmless" (to use a phrase from Hitchhiker's Guide to the Galaxy) characteristics of some special (Real, Complex) special cases that are mostly irrelevant from a category theory viewpoint ?

Tony

This is called Bott periodicity, and goes far beyond Clifford algebras.

Tony, let me expand on that comment.

The Clifford periodicity and all the classical group stuff presumes a concrete definition of the Reals or Complex numbers. In contrast, I am working on a constructive number theory, in which the Complex numbers should have an entirely new definition. Then there should be something generalising Bott periodicity.

I guess my question should be

how do things like Bott periodicity

and Hopf fibrations emerge from category theory?

Tony

There is a close connection between homotopy theory and category theory, at least amongst hard core mathematicians.

Tony, I took a class or maybe more from Hardy Mayer at UCI back in the early 1980s. Our secret technique to slow down his lectures was to ask him to do a small arithmetic calculation. Something a little harder than 2+3 = 5. This would eat up the rest of the lecture time as he was a complete disaster at actual calculations. He was in Europe during the excitement with Hitler (Romania?) and learned electronics from scavenging aircraft wrecks.

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