# 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.

74 questions
0answers
26 views

### Java Library that supports Quaternions Octonions, Sedenions?

I would like to experiment with multi dimensional complex numbers such as quaternions octonions, sedenions. I know Apache Commons Maths supports Quaternions, and I've found (although cannot download) ...
0answers
24 views

### Hermitian Octonion matrices?

The space of $n\times n$ octionion matrics $A$ such that $AA^*=A^*A=I_n$ is not a Lie group due to lack of associativity. But is it a smooth manifold? What is its dimension?
0answers
15 views

0answers
56 views

### The order of elements in finite octonions

After great success of this question I would like to continue this thread. Choose any element $x$ in algebra $\mathbb O_q$ of octonions over finite field $\mathbb F_q$. There is following equation ...
1answer
216 views

### What is a good book on general octonion algebras and the Cayley-Dickson construction?

I want a good reference that discusses properties of octonion algebras, especially over number fields. I'd like to know more about how this generalizes when applying the Cayley-Dickson construction ...
1answer
74 views

### Why is not possible to show that $S^7$ is a Lie Group in the following way?

I am taking a course about smooth manifolds following Elon Lages Lima's "Variedades Diferenciáveis" with Lee's "Introduction to smooth manifolds" (loosely following as a supplement) and I've stumbled ...
1answer
117 views

### Is it possible to plug hypercomplex numbers into the Riemann Zeta function?

I'm aware of the more detailed question: How to raise a number to a quaternion power However, from a more high-level perspective (read: probably less mathematical, hence my choice for raising ...
1answer
76 views

### Unital nonalternative real division algebras of dimension 8

The finite-dimension division algebras over the reals are: $\Bbb R$: the reals (dimension 1) $\Bbb C$: the complex numbers (dimension 2) $\Bbb H$: the quaternions (dimension 4) $\Bbb O$: the ...
1answer
118 views

### How does linear algebra over the octonions and other division algebras work?

An interesting question, which has been discussed in many forms on this site, is how many results from the study of linear algebra over vector spaces carries over when we allow the scalars to form an ...
1answer
86 views

1answer
60 views

1answer
280 views

### Do the octonions form a field?

The octonions are a noncommutative nonassociative normed division algebra over $\mathbb{R}$. Multiplication distributes over addition. Somehow, the existence of a norm implies the existence of ...
5answers
1k views

2answers
200 views