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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89 views

Fundamental theorem of algebra for Quaternions and Octonions — and the completeness

The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex $\mathbb{C}$ coefficients has at least one complex root. This includes polynomials with real ...
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Is there a reasonable limit to how far you can generalise complex numbers? [duplicate]

Real numbers satisfy a(bc) = (ab)c as well as ab = ba. They are also comparable. Generalising to complex numbers, everything stays the same, except the numbers lose their comparibility. Generalising ...
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Can octonions be represented by infinite matrices?

It is sometimes possible to multiply matrices of countably-infinite dimension. (Matrix multiplication is defined in the usual way, with rows and columns multiplied termwise and summed.) However, it ...
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49 views

Why this is the octonions? [closed]

The multiplication table is different from that of the octonions what I know. I tried to find suitable basis change for which the multiplication table coincide with wikipedia, but no success.
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Quaternionic and octonionic analogues of the Basel problem

It is a well-known fact that $$\sum_{0\neq n\in\mathbb{Z}} \frac{1}{n^k} = r_k (2\pi)^k$$ for any integer $k>1$, where $r_k$ are rational numbers which can be given explicitly in terms of Bernoulli ...
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Interpolation Scheme Between a set of Quaternions

My question is similar to Quaternion barycentric interpolation, but since that question is 6+ years old, I figured it could be asked again. I also will eventually be trying to do this with unit ...
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1answer
57 views

Proving that the Octonion Norm Respects Multiplication

I'm working with the following definition of the octonions: $\mathbb{O} = \mathbb{H} \times \mathbb{H}$, endowed with the product $$(p,q)(r,s) = (pr - sq^*, p^*s + rq).$$ Conjugation is given by $(p,q)...
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83 views

Properties lost when going from real number system to quaternions and octonions

What properties do we lose as we go from real numbers to quaternions, then to octonions? Do any new properties arise, or do calculations just become more "path dependant"?
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Is there a classification of bivector-valued cross products?

Background Vector-valued cross products Let $\mathbb{F}$ be an field of characteristic $0$. A $k$-ary cross product in the vector space $V=\mathbb{F}^n$ equipped with an inner product $\langle \cdot, \...
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A doubt from book Spinors and Calibrations

In chapter 14, the author shows that the octonionic projective plane is the quotient of exceptional Lie group $F_{4}$ by the group $Spin(9)$ (Theorem 14.99). When he is leading with an element that ...
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144 views

who pioneered the study of the sedenions?

The nature of this question is pure historical curiosity. I found lots of background information about the discovery of both Imaginary and Complex Numbers, and enough information about the first two ...
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Normed Division Algebras: Why those dimensions? [duplicate]

I recently read a fascinating article about string theory, which discussed higher-dimensional algebras and their applications to supersymmery. The author mentioned that there were only four algebras ...
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1answer
68 views

An octonion generalization of the usual quantum angular momentum operators.

I'm working through, developing the formalism for an octonion generalization of angular momentum operators, but the whole time I'm thinking that this has probably been done elsewhere. My searches ...
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Octonion Algebras Over Number Fields

Is there any textbook or paper about Arithmetic of Octonion Algebras or Octonion Algebras constructed over number fields? I know J. Voight book and K. Martin notes about quaternion algebras but I was ...
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461 views

Is the seven-dimensional cross product unique?

I'm confused about how many different 7D cross products there are. I'm defining a 7D cross product to be any bilinear map $V \times V \to V$ (where $V$ is the inner product space $\mathbb{R}^7$ ...
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Real, Complex, Quaternionic and Octonionic Projective spaces

Do Octonionic Projective spaces exist or defined similar to $\Bbb RP^n$, $\Bbb CP^n$, $\Bbb HP^n$? If so, are they symmetric spaces? I am asking this question because I've never seen Octonionic ...
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60 views

Why do $n$-dimensional numbers follow a geometric sequence?

I'm new here so please forgive my poor formatting and understanding. I'm trying to understand why $n$-dimensional numbers follow a sequence. For example, octonions are $8D$ numbers that (I assume) ...
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A real example of an Octonion product

Goal: find the general Octonion multiplication product like the Quaternion formula given here: https://en.wikipedia.org/wiki/Quaternion#Multiplication_of_basis_elements ...
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Subalgebras of the Real Octonions under Multiplication

I am interested to know what the subalgebras of the octonions look like. I tried searching for it but to no avail. What did show up was "Subalgebras of the Split Octonions," which was quite nice. Yet, ...
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1answer
44 views

Max number of solutions to $\exp\bigg(\frac{\rho}{\log(x)}\bigg)=x$ for complex, quaternion and octonion numbers?

If $\rho$ is a positive real number and $x$ is a real variable, I have verified that the following equation should have exactly two solutions, related to each other by being reciprocals of each other. ...
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1answer
76 views

7D octonionic cross product versus bivector cross product

Why the octonionic cross product product is not "unique" up to some permutations in the octonionic multiplication table BUT the bivector cross product in 7D is unique and isomorphic to one of those ...
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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 the ...
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Commutation or Anti-commutation of the corresponding imaginary units of the octonions and split-octonions

Given a general octonion x: $\mathbb{O}$=$\mathbb{H}$+$\mathbb{H}$$L$ by x=$x^1$+$x^2$i+$x^3$j+$x^4$k+$x^5$i$L$+$x^6$j$L$+$x^7$k$L$+$x^8$$L$ with $L^2$=-1, and a general split-octonion x: $\...
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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) ...
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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?
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Octonion Grassmanian?

I just learnt that the octonion projective space $\mathbf{OP}^n$ exists only for $n=1,2$. How about the "octonion Grassmannian", i.e. the space of $k$-subspace of $\mathbf O^n$? For which values of $(...
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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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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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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^...
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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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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 ...
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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 Diferenci√°veis" with Lee's "Introduction to smooth manifolds" (loosely following as a supplement) and I've stumbled ...
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215 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 ...
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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 ...
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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 ...
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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 ...
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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\...
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Is it possible to unify the number system? [closed]

And have a unified set of numbers? Now one can ask what the use of this is but let's leave that aside. I think Sedenions in a sense are the highest we have been up to. What if we were to go beyond ...
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Is octonion structure related to the fact that an 8-cube has an equal number of 2-faces and 3-cells?

Background: We know that octonions exist in 8-space, and we know that in 8-space, the 8-dimensional "measure polytope" ("hypercube") just so happens to have the SAME number of 2-dimensional faces (...
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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?
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$\mathbb{S}^1$-action and octonionic multiplication can be associated

Let $\mathbb{S}^7$ be the unit sphere of $\mathbb{R}^8$, which can be identified with the unit octonions. The circle $\mathbb{S}^1$ naturally acts on $\mathbb{S}^7$ by complex multiplication: $$z \...
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Is $ SL_2(\mathbb{O})$ a Lie group?

In JOhn Baez's article The Octonions, he writes the lie algebra isomorphism $\mathfrak sl_2(\mathbb {O}) \space \tilde = \space \mathfrak so(9, 1) $. Why is the former a Lie algbra, even though $SL_2(...
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1answer
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Mirror symmetries of the Albert algebra

A simple Euclidean Jordan algebra (i.e. a factor) is either a spin-factor, the matrices over the reals/complex-numbers/quaternions or the exceptional Albert algebra of 3x3 octonian matrices. My ...
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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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248 views

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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Is there a terminology that generalizes the “real part” and the “imaginary part”?

By the Cayley-Dickson construction quaternions can be written as $q=z+wj$ and octonions can be written as $x=q_1+q_2l$. Is there any generalization of the real part and the imaginary part such that $$\...
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1answer
442 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 ...
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187 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 $...