# Tagged Questions

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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### Octonionic formula for the ternary eight-dimensional cross product

A cross product is a multilinear map $X(v_1,\cdots,v_r)$ on a $d$-dimensional oriented inner product space $V$ for which (i) $\langle X(v_1,\cdots,v_r),w\rangle$ is alternating in $v_1,\cdots,v_r,w$ ...
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### Generators of so(7)

Short version: Let $V$ be a 7-dimensional linear space of (real) square matrices. Suppose further that $[V,V]$ (the linear space spanned $[X,Y]$, $X,Y\in V$) is isomorphic to $\mathfrak{so}(7)$. Can ...
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### Sextonion Cayley Table

I've been reading up on the sextonions and was wondering if it would be possible to construct a Cayley table for the split sextonions the same way as one would do so for the split quaternions and ...
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### Why do division algebras always have a number of dimensions which is a power of $2$?

Why do number systems always have a number of dimensions which is a power of $2$? Real numbers: $2^0 = 1$ dimension. Complex numbers: $2^1 = 2$ dimensions. Quaternions: $2^2 = 4$ dimensions. ...
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### 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 ...
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### Higher dimensional cross product

I know that cross products do not exist in 4, 5 or 6 dimensions, but do in 7 dimensions. So I was wondering if this was because cross products can be considered the imaginary part of $2^n - ion$ ...
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### What is the use of sets above the Complex set?

I recently started reading about sets above the complex set (the set of quaternions, the set of octonions, etc...) and since I already had a lot of difficulty understanding why complex numbers were ...
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### Sum of eight squares over a finite field.

Consider the split-octonions $\mathbb{O}$ with coefficients in $\mathbb{F}_q$. Suppose $a \in \mathbb{F}_q$ and $b \in \mathbb{F}_q^*$. I want to find the amount of elements $x \in \mathbb{O}$ such ...
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### Using the Fano plane for octonion multiplication

The Fano plane is the projective plane over the field $\mathbf Z/2$. It can be used to remember octonion multiplication, as nicely explianed in John Baez's article on octonions (see http://math.ucr....
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### Motivating the Cayley-Dickson construction by proving Hurwitz's theorem

To me it seems the way to motivate the Cayley-Dickson construction is to prove Hurwitz's theorem, which is done over at Wikipedia. The theorem states the only real division algebras equipped with a ...
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### Polarization identity $2(a,b)(c,d)=(ac,bd)+(ad,bc)$

I am interested in following along this Wikipedia article's derivation of properties of composition algebras (in particular, Euclidean Hurwitz algebras). Let $A$ be a unital, not necessarily ...
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### Can a point have a nontrivial isometry group?

This question is extremely related to this other question. In fact, a positive answer here directly implies a positive answer there. However, since it is a mathematically different question I decided ...
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### Has the Riemann Hypothesis been generalized to the Octonions and the Quaternions?

I've noticed that it uses imaginary numbers. I know that sometimes when I have too few dimensions like (-1)^n, dots show where I might expect lines due to imaginary numbers. So perhaps there is a ...
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### Inner product space over generalized number systems

Apologies for the lengthy setup, but I want to make sure I am clear on how I am using the notation, and what I mean by the phrase "generalized number system". Define a generalized number system $G$ ...
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### Construction of Hyper-Complex Numbers

How does one construct a hyper-complex number multiplication table? For example: Quarternions: ...
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In the wikipedia atrticle (http://en.wikipedia.org/wiki/Octonion) it is stated that "one can show that the subalgebra generated by any two elements of $\mathbb{O}$ is isomorphic to $\mathbb{R}$, $\... 1answer 63 views ### Unit or non-zero octonions form an$A_\infty$-space? If I have a Moufang loop, can it have a classifying space? I'm thinking of the unit octonions, if that's too general, so perhaps a better question is: are the unit (or non-zero) octonions an$A_\infty$... 1answer 184 views ### Are roots of unity in hypercomplex algebras well defined? While playing around with cyclotomic fields, I started to wonder about taking the roots of unity in higher dimensional analogues of the complex plane. Are the roots of unity well defined in the ... 2answers 144 views ### What can you do with octonions? How can you calculate with them and what can you actually make up from the calculations? And what is exactly meant by normed division-algebras? 1answer 379 views ### What specific algebraic properties are broken at each Cayley-Dickson stage beyond octonions? I'm starting to come around to an understanding of hypercomplex numbers, and I'm particularly fascinated by the fact that certain algebraic properties are broken as we move through each of the$2^n$... 1answer 196 views ### Why do we start losing algebraic properties when dealing with hypercomplex numbers? [duplicate] Every form of hypercomplex number I have seen (including the complex numbers) lose some important algebraic property. Why is that? Is there a pattern to what we lose? 2answers 422 views ### Trinonions, Quaternions, Quinonions, Sextonions, Septonions, Octonions There are quaternions and octonions and even sextonions but what about trinonions, quinonions and septonions. Are there 3, 5, and 7 dimensional algebras which could be called trinonions, quinonions ... 1answer 99 views ### Pfister's 16-Square Identity and the norm of sedenions Consider the sequence of numbers: complex numbers$\Bbb C$, quaternions$\Bbb H$, octonions$\Bbb O$, and sedenions$\Bbb S$. The Brahmagupta-Fibonacci 2-Square identity implies that the norm of the ... 1answer 39 views ### Are H$_n$(O) (n>3) Jordan algebras? As we know, H$_3$(O) is a 27-dimensional exceptional Jordan algebra, here O is Cayley octonion algebra.But how about n>3? I guess that when n>3, H$_n$(O) are not Jordan algebras. But I only have a ... 0answers 201 views ### Fractal derivative of complex order and beyond Is there some precise definition of "complex (fractal) order derivative" for all complex number? I am aware of the Riemann-Liouville fractional definition given here: Complex derivative but I would ... 3answers 317 views ### Math beyond Quaternions Quaternions remove the commutative property and octonions eliminate the associative property can we go any higher and eliminate more properties? 2answers 1k views ### What do we lose passing from the reals to the complex numbers? As normed division algebras, when we go from the complex numbers to the quaternions, we lose commutativity. Moving on to the octonions, we lose associativity. Is there some analogous property that we ... 0answers 82 views ### A question about hypercomplex numbers: quaternions, octonions etc [duplicate] Possible Duplicate: Why is 8 so special? First of all let me state that I am not a mathematician but I work in computer science and engineering. I was reading about hypercomplex numbers, and ... 1answer 200 views ### Constructing a parallelization of the 7-sphere. I would like to show that$S^7$, the 7-sphere, is a parallelizable manifold. Let$\mathcal{O}$be the octonions, the normed division algebra (noncommutative, nonassociative) over$H\times H$, where$H$... 2answers 1k views ### Why are properties lost in the Cayley–Dickson construction? Motivating question: What lies beyond the Sedenions? I'm aware that one can construct a hierarchy of number systems via the Cayley–Dickson process: $$\mathbb{R} \subset \mathbb{C} \subset \mathbb{H}... 4answers 635 views ### Matrix Representation of Octonions Since quaternions \mathbb{H} have a matrix representation as elements of \text{SU}(2,\mathbb{C}) as the following$$ 1 \mapsto \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix},\quad \mathrm ... 1answer 1k views ### Why is 8 so special? I have been reading about multi-dimensional numbers, and found out that it's been proven that the Octonions are the composition algebra of the largest dimension. I was wondering why, despite having ... 1answer 325 views ### Octonionic Hopf fibration and$\mathbb HP^3$Real, complex and quaternionic versions of Hopf fibration ($S^0\to S^n\to\mathbb RP^n$,$S^1\to S^{2n+1}\to\mathbb CP^n$and$S^3\to S^{4n+3}\to\mathbb HP^n$) give rise to spherical fibrations$S^1\to\...
While reading about Moufang loops in the book "An introduction to Quasigroups and their Representations" by Smith, I've encountered the following statement: The set $S^7$ of nonzero octonions of ...