Why are properties lost in the Cayley–Dickson construction?

Question

Motivating question: What lies beyond the Sedenions?

I'm aware that one can construct a hierarchy of number systems via the Cayley–Dickson process:

$$\mathbb{R} \subset \mathbb{C} \subset \mathbb{H} \subset \mathbb{O} \subset \mathbb{S} \subset \ldots $$

"Reals" $\subset$ "Complex" $\subset$ "Quaternions" $\subset$ "Octonions" $\subset$ "Sedenions" $\subset$ $\ldots$

and that at each step you're given a multiplication table that tell how the elements interact. As you move up the ladder, certain "nice" properties are lost: ordering, commutativity, associativity, multiplicative normedness, etc... Given the multiplication table, you can show that these properties don't hold.

Eric Naslund noted that "the first 4 are very special as they are the unique 4 normed division algebras over ℝ", no surprise then that these $2^n$-ions have found quite a bit of use. I'm interested in the sequence itself however, irrespective of how useful a $2^{256}$-ion might be (ducenti-quinquaginta-sex-ion?).

I feel like something deeper is going on here though that I don't understand. Why are these particular properties lost at each step? Is it possible to quantify the process such that, at the $2^n$-ion you can say something about the symmetry of the multiplication table*?

* I'm making an ansatz that there is a connection between the symmetry of the multiplication table and these "nice" properties.

Answer 1

There are eight equivalent definitions of the Cayley-Dickson product of ordered pairs, but one that is commonly used is $(a,b)(c,d)=(ac-db^*,a^*d+cb)$ where the conjugate (for all eight variations) is defined as $(a,b)^*=(a^*,-b)$.

The unit basis vectors $e_0,e_1,e_2,\cdots$ for all finite dimensional Cayley-Dickson vectors may be defined in various ways, but to preserve the inherent symmetry of the multiplication table, the most natural way to define the sequence is $e_{2n}=(e_n,0)$ and $e_{2n+1}=(0,e_n)$ for $n\ge0$. Note that this produces a different numbering from the usual numbering of the Octonion basis vectors as used by Octonion specialists and, that as a result, their multiplication table does not reveal the inherent symmetry of the process.

In addition to revealing the symmetry of the Cayley-Dickson process, this numbering of the basis vectors has the added advantage that for all non-negative integers $i,j$ it is true that $e_ie_j=\pm e_k$ where $k$ is the bit-wise "exclusive or" of the binary representations of $i$ and $j$.

Using this numbering of the basis vectors, the multiplication table of any finite dimensional Cayley-Dickson space can be recovered from the following properties:

1. $e_1e_{2n}=e_{2n+1}$ for all $n\ge0$.
2. If $0\ne i\ne j\ne 0$ then $e_ie_j=e_k$ implies all the following:
   1. $i\ne k$
   2. $j\ne k$
   3. $k\ne 0$
   4. $e_je_i=-e_k$
   5. $e_je_k=e_i$ (these last two are the quaternion properties)
   6. $e_{2i}e_{2j}=e_{2k}$
   7. $e_{2i}e_{2j+1}=e_{2k+1}$
   8. $e_{2i+1}e_{2j}=e_{2k+1}$
   9. $e_{2j+1}e_{2i+1}=e_{2k}$ (note the reversal of $i$ and $j$)

For four of the other seven alternate ways to define the Cayley-Dickson product, property 1 differs, and for all seven, some of the rules 2.7, 2.8 and 2.9 differ.

Now whether the symmetry of the multiplication tables will explain the loss of properties, I do not know, but it might be an interesting subject for someone to investigate.

More information and links to my research can be found on my webpage at http://jwbales.us/