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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30
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2answers
2k 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 ...
28
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2answers
2k 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 $2^n$ ...
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5answers
7k views

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 ...
22
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2answers
3k 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}...
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3answers
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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 ...
12
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1answer
2k 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 ...
11
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0answers
221 views

Does the exceptional Lie algebra $\mathfrak{g}_2$ arise from the isometry group of any projective space?

I learned from Baez's notes on octonions that the classical simple Lie algebras can be identified with the Lie algebras of isometry groups of projective spaces over $\mathbb{R}, \mathbb{C}$ and $\...
9
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1answer
207 views

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 $v_1,\cdots,v_r,w$ ...
8
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2answers
859 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 ...
8
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387 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 ...
7
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5answers
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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 ...
7
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2answers
263 views

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. ...
7
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1answer
112 views

Do the octonions contain infinitely many copies of the quaternions?

Note: by "infinitely many", I'm confident I always mean $\beth_1$ many herein. We can easily show the quaternions contain infinitely many copies of $\Bbb C$ because, given any unit vector $\in\Bbb R^...
7
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1answer
510 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 ...
6
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1answer
184 views

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 ...
6
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213 views

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 ...
6
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0answers
158 views

Are complex split-octonions isomorphic to a more easily-defined algebra?

I write fiction and nonfiction, both which are mathy. My fiction is not usually supermathy but I'm working on a fictional story that has some math in it, and I prefer accuracy to mathbabble. I'm ...
5
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1answer
742 views

Can octonions be used to rotate 7-dimensional vectors?

A friend told me that, the same way you can represent a 3-vector as an imaginary quaternion then conjugate it by a unit quaternion with real part $\cos(\frac{\theta}{2})$ to rotate it by $\theta$ ...
5
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1answer
140 views

Is there a slick proof for the identity that expresses the inner product of imaginary octonions in terms of the cross product?

Consider the octonions $\mathbb O$ and in particular their imaginary part $\operatorname{Im}\mathbb O$. Let $(-,-)$ be the scalar product induced by the identification of the imaginary octonions with $...
5
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1answer
263 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 ...
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3answers
521 views

Math beyond Quaternions

Quaternions remove the commutative property and octonions eliminate the associative property can we go any higher and eliminate more properties?
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3answers
273 views

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 $G$ ...
4
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1answer
467 views

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....
4
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1answer
312 views

Proof that the subalgebra generated by any two elements of $\mathbb{O}$ is isomorphic to $\mathbb{R}$, $\mathbb{C}$, or $\mathbb{H}$

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 $\mathbb{R}$, $\...
4
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1answer
465 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\...
4
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1answer
123 views

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 ...
3
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2answers
413 views

Nonzero Octonions as a 7-sphere

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 ...
3
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2answers
120 views

Understanding terminology of, fibers, clutchings and Hopf.

I have some questions regarding the terminology of fiber bundles as used in section 3 of this paper; http://www.sciencedirect.com/science/article/pii/S0723086907000151 The section starts off by ...
3
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1answer
311 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 ...
3
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1answer
134 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 ...
3
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1answer
84 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 $A_\infty$...
3
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1answer
229 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 ...
3
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1answer
99 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 ...
3
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1answer
317 views

Meaning of this exclamation mark?

In section 3 of the paper https://www.sciencedirect.com/science/article/pii/S0723086907000151 The author constructs a fiber bundle $(\rho_n)\zeta$ by taking the pullback of the diagram $S^8\...
3
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0answers
46 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) ...
3
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0answers
47 views

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 ...
3
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0answers
61 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 ...
2
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1answer
210 views

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 $2^n - ion$ ...
2
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2answers
164 views

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 ...
2
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1answer
89 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 ...
2
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1answer
157 views

Algebra Symbols $\mathfrak h$ and $\mathfrak{so}$

What do these symbols mean in algebra? I found them as follows: $$\mathfrak h_3(\Bbb O(\Bbb Z_p))$$ $$\mathfrak{so}(\Bbb O)\oplus\Bbb O^3$$
2
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1answer
121 views

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 ...
2
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1answer
134 views

The geometric interpretation of quaternion and Octanion

Can anybody give me any useful link for the history of quaternion? The quaternion and Octanion are constructed but why other do not exist? What is the geometric interpretation of a quaternion?
2
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1answer
105 views

Are there useful algebras between the Cayley-Dickson algebras?

Reals, complex numbers, quaternions, octonions, etc are a hierarchy of algebras which can be constructed in a regular way. One obvious property of this hierarchy is that each such algebra has $2^n$ ...
2
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2answers
108 views

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 ...
2
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1answer
53 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 ...
2
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0answers
71 views

Construction of the Octonions

I get how to construct number systems: $\Bbb{N,~Z,~Q,~R,~C}$. But what about the hypercomplex? How do we go from $\Bbb C$ to $\Bbb H$ and $\Bbb O$? I've also heard it stated that the octonions are ...
2
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0answers
29 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?
2
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0answers
22 views

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 ...
2
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1answer
65 views

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 $2\times 2$ ...