# 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.

75 questions
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### The isotropy subgroup of the action of Spin(7) on the Grassmannian G(3,8)

The group Spin$\left( 7\right) \$is the universal cover of $SO\left( 7\right)$ and is characterized as the subgroup of $SO\left( 8\right)$ consisting of the automorphisms of the triple cross ...
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### Classification of subalgebras of composition algebras

Let $F$ be an algebraically closed field. It is known that the only composition algebras over $F$ are $F$ itself, the direct sum $F\oplus F$ (also called split-complexes), the algebra of $2\times 2$ ...
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### If there are any 3nion, 5nion, 7nion, 9nion, 10nion, etc.

The quaternion/octonion extend the complex numbers, which extend the real numbers. So we go: 1-tuple: Real numbers. 2-tuple: Complex numbers. 4-tuple: Quaternions. 8-tuple: Octonions. The Wikipedia ...
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### If there can exist a model of the octonions without complex numbers

I know the ordering of complex numbers -> quaternions -> octonions, so the octonions are basically built on top of the complex numbers. But I am wondering, still, if there is a way to reformulate them ...
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### Is there a way to introduce quaternions and octonions in a similar way to how we are typically introduced to complex numbers?

So I've been reading a little bit into ideas around quaternions and octonions. I just read the following explanation that introduces them as what happens when you have complex numbers and you then ask ...
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### Characterizing lattices in $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ and $\mathbb{O}$ that are also rings

I am trying to find necessary and sufficient conditions for a nondegenerate lattice in one of the real division algebras $\mathbb{K}$ to admit the structure of a ring with identity (alternative ...
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### Why is SO(8) generated by left multiplications by octonions?

I have read that $SO(8)\simeq SO(\mathbb{O})$ is generated by the set $\{L_a \,|\, a\in S^7\}$, where $L_a:\mathbb{O}\to\mathbb{O}, \, x\mapsto ax$ is the left translation. Since for $a\in\mathbb{O}$,...
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### What are some real-world uses of Octonions?

... octonions are the crazy old uncle nobody lets out of the attic: they are nonassociative. Comes from a a quote by John Baez. Clearly, the sucessor to quaterions from the Cayley-Dickson process is ...
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### Octonions - affine space

I'm writing a project on Cayley's algebra. I have some topics which I have to follow and I've managed to solve most of them,except 2. I have written about their rule of multiplication,together with ...