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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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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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78 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^...
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1answer
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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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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 ...
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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 ...
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1answer
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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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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 ...
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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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Since the Fano plane captures properties of octonions, does it capture properties of the $E_8$ root-system [closed]

Since the Fano plane captures certain properties of octonions does it capture certain properties of the $E_8$ root-system? If so, could you describe how, or provide links to explanations and ...
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1answer
312 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\...
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2answers
116 views

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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1answer
126 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?
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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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1answer
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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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1answer
118 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 ...
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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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1answer
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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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1answer
86 views

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
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 ...
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1answer
131 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 $...
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1answer
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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 $\theta$ ...
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1answer
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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 $2^n$ ...
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131 views

Can one test an octonionic interpretation for a conjecture, apparently valid in the complex and quaternionic settings, and proven in the real case?

For the values $\alpha = \frac{1}{2},1, 2$, corresponding to real, complex and quaternionic scenarios, the formulas (https://arxiv.org/abs/1301.6617, eqs. (1)-(3)) \begin{equation} \label{Hou1} P_1(\...
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How can we express determinant in terms of trace

The determinant of a matrix in $\mathfrak h_3(\Bbb O)$ is defined by $(a,b,c,\mathbf{a},\mathbf{b},\mathbf{c})=\begin{bmatrix} a &\mathbf{c} &\mathbf{b} \\ \mathbf{\bar{c}} & b & \...
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1answer
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Algebra Symbols $\mathfrak h$ and $\mathfrak{so}$

What do these symbols mean in algebra? I found them as follows: $$\mathfrak h_3(\Bbb O(\Bbb Z_p))$$ $$\mathfrak{so}(\Bbb O)\oplus\Bbb O^3$$
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1answer
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Do parallelisability of the 3- and 7-spheres arise out of the spheres themselves or an instrinsic property of the space in which they're embedded?

The 7-sphere is the highest dimensional sphere that is parallelisable, and the next highest is the 3-sphere. An inherent property of each of these spheres is that they're embedded in a space of ...
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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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1answer
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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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2answers
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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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1answer
189 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 $2^n - ion$ ...
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2answers
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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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1answer
115 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 ...
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1answer
428 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....
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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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1answer
74 views

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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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 ...
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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 $\...
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1answer
122 views

Is seven-dimensional cross product rotationally invariant?

For three-dimensional cross product, the following property holds true: \begin{equation} (R\mathbf x) \times (R \mathbf y)=R(\mathbf x \times \mathbf y) \end{equation} where $R\in SO(3)$. Is the ...
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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 ...
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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 ...