# Questions tagged [octonions]

For questions on the octonions, a normed division algebra over the real numbers. It is a non-associative higher-dimensional analogue in the hierarchy of real, complex, and quaternionic numbers.

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### Norm of the Sedenions

Let the Cayley-Dickson doubling of the octonions be called the sedenions. The sedenions are not a division algebra, because they contain zero divisors. The presence of zero divisors means that the ...
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Given $a\in \mathbb R^3-\{0\}$, then the map $J_a: q\to q\frac{a}{|a|}$ is a complex structure of the quaternions $\mathbb H$. Now, what about the octonions $\mathbb O$? Do they similarly obtain a ...
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### Orders of Paige Loops over Finite Fields

A Moufang loop $M$ is a loop that satisfies the Moufang identity: $(zx)(yz) = z((xy)z),\forall x,y,z\in M$. From here, we get the following statement: For any field $F$ let $M(F)$ denote the Moufang ...
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### Investigating and Generalizing Octonionic Nonassociativity

A possible multiplication convention for the octonions is given by the following 7 sets of integers from 1 to 7. {1,2,3},{1,4,5},{1,6,7},{2,4,6},{2,5,7},{3,4,7},{3,5,6}. Notice that each of the seven ...
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### Automorphism group of octonions is closed in $O(7)$.

So far I have understood that the group of automorphisms of the octonions is a subgroup of $O(7)$. I want to understand how it is a Lie group using the lemma that closed subgroups of Lie groups again ...
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### "Canonical" norm on a real finite-dimensional unital associative division algebra?

Let $\mathcal{C}$ denote the category of unital associative finite-dimensional division $\mathbb{R}$-algebras. (As a full subcategory of that of unital $\mathbb{R}$-algebras.) For $A\in \mathcal{C}$ ...
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### General algebra for rotations in $n$ dimensions

Rotations in $2$ dimensional space can be given a nice representation with complex numbers. The group $(\mathbb{C}, \times)$ under complex multiplication has two one-parameter subgroups: the real line ...
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