# What lies beyond the Sedenions

In the construction of types of numbers, we have the following sequence:

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

or:

$$2^0 \mathrm{-ions} \subset 2^1 \mathrm{-ions} \subset 2^2 \mathrm{-ions} \subset 2^3 \mathrm{-ions} \subset 2^4 \mathrm{-ions}$$

or:

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

With the following "properties":

• From $\mathbb{R}$ to $\mathbb{C}$ you gain "algebraic-closure"-ness (but you throw away ordering).
• From $\mathbb{C}$ to $\mathbb{H}$ we throw away commutativity.
• From $\mathbb{H}$ to $\mathbb{O}$ we throw away associativity.
• From $\mathbb{O}$ to $\mathbb{S}$ we throw away multiplicative normedness.

The question is, what lies on the right side of $\mathbb{S}$, and what do you lose when you go from $\mathbb{S}$ to one of these objects ?

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One useful generalization starting from $\mathbb{H}$ and extending to all powers of $2$ is Clifford algebras: en.wikipedia.org/wiki/Clifford_algebra . You can also keep applying the Cayley-Dickson construction past $\mathbb{S}$ (en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction) although I don't know if this is useful. – Qiaochu Yuan Nov 28 '11 at 17:13
Dragons. Beyond sedenions there are dragons. Beware. – Mariano Suárez-Alvarez Nov 28 '11 at 17:15
If you want to keep with your "throw away" theme, then from $\mathbb{R}$ to $\mathbb{C}$ you lose the ordering – Jason DeVito Nov 28 '11 at 19:23
(somewhat) related: mathoverflow.net/questions/19929/19975#19975 – Grigory M Nov 28 '11 at 23:02
@Jason: Thanks, I have add this to the first action from $\mathbb{R}$ to $\mathbb{C}$) – Willem Noorduin Nov 29 '11 at 9:26
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Remark: I am left wondering what is gained by going past Octonions. The the first 4 are very special as they are the unique 4 normed divison algebras over $\mathbb{R}$. Perhaps someone with more knowledge can point out the possible uses of the Sedenions and their higher counterparts.
As for applications of sedenions, I am quoting Wikipedia: "Moreno (1998) showed that the space of norm $1$ zero-divisors of the sedenions is homeomorphic to the compact form of the exceptional Lie group $G_2$." – darij grinberg Nov 28 '11 at 17:44