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Questions tagged [octonions]

Octonions form a 8-dimensional normed division algebra constructed over the reals. They can be seen as a non-associative (alternative) extension of the quaternions. They have been first defined and studied in the 19th century by John Graves and Arthur Cayley. There are several variants (such as split-octonions) and strong relations with Lie Groups and projective geometry.

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Borromean rings on $\Bbb{RP}^2$ and octonions

If I draw a trefoil knot on a projective plane and draw a circle around it touching the three outer parts of the curve. I can view this as a division of the projective plane in 8 triangles, viewed as ...
Maarten Havinga's user avatar
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Is there a "natural" interpretation of the power function for octonions and for sedenions?

This question is a sequel to Is there a definition of $\log(x)$ for quaternion/octonion $x$? Since $\log(x)$ is multivalued even for complex $x \in \mathbb{C}$, it is impossible to define $\log(x)$ ...
Dieter Kadelka's user avatar
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Is there a definition of $\log(x)$ for quaternion/octonion $x$?

I'm trying to implement $\log({\bf q})$ in python, where ${\bf q} = (q_0,\ldots,q_7) \in \mathbb{O}$ is an octonion. There is a well known definition of $\log({\bf q})$ for quaternions ${\bf q} = (s,v)...
Dieter Kadelka's user avatar
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318 views

Chirality of octonion algebras

Octonion multiplication can be defined with respect to a set of triads. A set of such triads can be represented by a directed Fano plane diagram such as the following two diagrams. This depicts two ...
John Wayland Bales's user avatar
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Four octonionic loops to identify

A loop is a quasigroup with the identity. I have to disclose that loop-theory is something outside my expertise. I have four loops arising from octonionic elements of unitary norm that have order 16, ...
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Classification of Moufang planes of real dimension 16

Incidence geometry is not really area of expertise so I'm asking here: are all Moufang planes of 16 dimension already classified? I'm not just interested in the compact ones. Is there already a ...
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Associating octonionic matrices

Is it possible for three square octonionic matrices to associate multiplicatively even though some or all of their entries do not? If so, is it possible to construct matrix groups in this way?
Daniel Sebald's user avatar
7 votes
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428 views

Quadratic forms on $\mathbb{R}^{16}$ coming from octonions

$\DeclareMathOperator\RRe{Re}\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\Sym{Sym}$Let $\mathcal{H}_2(\mathbb{O})$ denote the (10-dimensional) real vector space of octonionic Hermitian matrices ...
asv's user avatar
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p-adic analogue of octonions

There are the complex p-adic numbers. But what is the p-adic analogue of the Cayley–Dickson construction? Or more important: What is the p-adic analogue of the octonions? It would be nice if the (unit)...
Raoul's user avatar
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Diagonalization of octonionic Hermitian matrices of size $2\times 2$

The group $Spin(9)$ is a subgroup of $SO(16)$ and acts transitively on the unit sphere $S^{15}$. $Spin(9)$ acts naturally on the space of octonionic Hermitian $2\times 2$-matrices (I do not define ...
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Classification of octonionic reflection groups

I know that there exist classification theorems for real, complex, and quaternionic, reflection groups. There are presentations for the real reflection groups, as well as further presentations for the ...
Sean Miller's user avatar
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Reference for bi-octonionic projective plane $\left(\mathbb{C}\otimes\mathbb{O}\right)P^{2}$

I'm not well versed in projective geometry since it is not really my field. I read in [1] about the existance of a projective plane $\left(\mathbb{C}\otimes\mathbb{O}\right)P^{2}$ defined on bi-...
Dac0's user avatar
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Why is the automorphism group of the octonions $G_2$ instead of $SO(7)$

I can calculate the derivation of the octonions and I clearly find the 14 generators that form the algebra of $G_2$. However, when I do the same calculation for the quaternions, I end up with the ...
p6majo's user avatar
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Coordinate-free description of an alternating trilinear form on pure octonions

Let $O$ denote the division algebra of octonions over $\Bbb R$, and write $V$ for the 7-dimensional quotient space $O/{\Bbb R}$. The compact group $G_2:={\rm Aut}(O)$ naturally acts on $V$, and ...
Mikhail Borovoi's user avatar
12 votes
1 answer
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Unit group of octonions over finite fields

One can define the algebra $A(K)$ of octonions over an arbitrary field $K$, see for example the command OctaveAlgebra in GAP: https://www.gap-system.org/Manuals/doc/ref/chap62.html . When $K$ is a ...
Mare's user avatar
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Is the average associator over a finite subloop of octonions necessarily zero?

For any three octonions $a,b,c$, their associator is defined as \begin{equation*} [a,b,c]=a(bc)-(ab)c \end{equation*} and measures their non-associativity so to speak. Now suppose that $L$ is a finite ...
Bob's user avatar
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Automorphism group of formally real Jordan algebras of hermitian matrices

It is well known that the automorphism group of exceptional Jordan algebra $\mathcal{h}_{3}(\mathbb{O})$ is the exceptional Lie group $F_{4}$. I am trying to understand the automorphism group of the ...
Inácio 's user avatar
12 votes
1 answer
376 views

Characteristic polynomial of an $8 \times 8$ symmetric matrix with indeterminate entries related to octonionic multiplication

I consider $1,i,j,k,l,m,n,o$ the standard basis of the (complexified if you like) octonions ($\mathbb{O}$ for the octonions). Let $a = x_1.1 +\ldots + x_8.o$, $b = x_9.1+ \ldots + x_{16}.o$ and $c = ...
Libli's user avatar
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Octonion algebras over number fields [closed]

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 ...
N Brasilis's user avatar
1 vote
2 answers
174 views

Determinants in Jordan algebras of Euclidean type

As far as I heard (I am not sure about the precise statement) there is a classification of simple Jordan algebras of Euclidean type. Each (some?) of such algebras admits a cone of positive definite ...
asv's user avatar
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6 votes
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Unital nonalternative real division algebras of dimension 8

Cross-posted from Math SE because I felt like it might be too obscure for there. I'm sorry if this is the wrong place for it. EDIT: This question now has an answer over there The finite-dimension ...
Akiva Weinberger's user avatar
17 votes
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When is the determinant an $8$-th power?

I am working over $\mathbb{R}$ (though most of the story goes over any field). I am looking for linear spaces of matrices such that the restriction of the determinant to this spaces can be written (...
Libli's user avatar
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22 votes
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What finite simple groups we can obtain using octonions?

Rearranged on 2017-05-31 What I am missing is a uniform definition of finite simple groups. Especially sporadic groups are difficult to define. Many of the finite groups are defined using machinery ...
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115 views

A general form of a maximal totally isotropic subspace in the split octonion algebra

Let $\mathbb O'$ be the split octonion algebra over $\mathbb R$. For each nonzero divisor of zero $x\in \mathbb O'$ $\mbox{($x \neq 0, N(x)=0$)}$ the kernel of the left multiplication by $x$, $Ker ...
Yourij1's user avatar
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2 answers
456 views

Are two octonion algebras with different multiplications isomorphic?

Some authors, e.g. Baez, Ward, defined multiplication of octonions by formula $ (a,b) \cdot^B (c,d)=(ac-db^*, cb+a^*d) \textrm{ for } a,b,c,d\in \mathbb H, $ some others, e.g. Springer & Veldkamp,...
M.M's user avatar
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1 answer
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A general form of mappings preserving angle between vectors and their image in $R^8$

In $\mathbb R^8$, identified with the octonion algebra $\mathbb O$, mappings $f: O \rightarrow \mathbb O$ of the form $x \mapsto xu $ and $ x \mapsto ux $, where $u$ is a fixed unit octonion (i.e. ...
user 200's user avatar
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277 views

Matrix representation of the automorphisms of the octonion's algebra without Lie's theory

It is known that if a function $g$ is an automorphism of the algebra of octonions then there is an orthogonal basis of a form: $1,e_1, e_2, e_3=e_1e_2, e_4, e_5=e_1e_4, e_6=e_2e_4, e_7=e_3e_4$, where ...
user125's user avatar
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5 votes
2 answers
571 views

About some property of automorphism of octonions

Let $f$ be an automorphism of the algebra of octonions. Is it true that $f$ preserves some quaternionic subalgebra? Has the statement an elementary proof?
Yourij1's user avatar
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6 votes
2 answers
501 views

Constructing real forms of the Tits-Freudenthal magic square for (Rosenfeld) projective planes

If $\mathbb{K},\mathbb{L} \in \{\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O}\}$ then the Rosenfeld projective ("elliptic"?) plane $\mathbb{P}^2(\mathbb{K}\otimes\mathbb{L})$ is "the" compact Riemannian ...
Gro-Tsen's user avatar
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10 votes
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Atiyah on the "Galois group of the octonions" and Physics

Apparently Atiyah was talking about the "Galois group of the octonions" and the unification of the forces of physics at the Heidelberg Forum. Unfortunately not on the stage -- it didn't make its way ...
Trent's user avatar
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6 votes
1 answer
382 views

Properties of complexified octonions

I have following questions about the complexified octonion algebra $\mathbb C \otimes_{\mathbb R} \mathbb O$. Zero divisors are of shape $p+i\otimes q$ (shortly $p+iq$) where $p$, $q$ are ...
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1 vote
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171 views

Generating $\mathfrak{so}(7)$

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 actually a subalgebra isomorphic to $\...
Llohann's user avatar
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11 votes
1 answer
722 views

Determinants of octonionic hermitian matrices

For quaternionic hermitian matrices (i.e. quaternionic square matrices $(a_{ij})$ satisfying $a_{ji}=\bar a_{ij}$) there is a nice notion of (Moore) determinant which can be defined as follows. ...
asv's user avatar
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6 votes
0 answers
267 views

Exceptional symmetric spaces embedded in exceptional Lie group

In Yokota (1959) and Atsuyama (1977) papers one can find embedding of projective space $\mathbb OP^2$ into Lie group $F_4$. Lately I come to following idea to have embedding of all four projective ...
user avatar
3 votes
1 answer
138 views

Transitivity of $Spin(7)$ in triples of vectors

I have a simple question: transitivity of $Spin(7)$ in triples of orthogonal vectors. Let $Spin(7)\subset SO(8)$ act on $\mathbb{R}^8$, and $e_1,e_2,e_3$, $v_1,v_2,v_3$ be two triples of mutually ...
Jjm's user avatar
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9 votes
1 answer
376 views

Expressing $SO_8$ element as product of $L_u$ and $R_u$ for unit octonions $u$

Welcome octonions friends ! Long time ago when I travelled through octonion land, I conjectured that every $SO_8$ element can be expressed as product $L_a L_b R_c R_d$ for unit octonions $a$, $b$, $c$...
user avatar
6 votes
1 answer
221 views

Does the Cayley-Dickson construction preserve isomorphism of quaternion algebras?

I posted this on math.stackexchange to no avail, so I hope it's appropriate to post here despite that it might not be research-level. I expect the answer to this is well-known to people studying non-...
j0equ1nn's user avatar
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5 votes
1 answer
307 views

Looking for Severi varieties

Let $K$ be an algebraically closed field of characteristic $0$, and let $\mathbb{O}$ be the Cayley algebra over $K$. Let $$ \mathfrak{J}_{3}=\{A\in\mathcal{M}_{3}(\mathbb{O}):A\text{ is Hermitian}\}, ...
Srinivasa Granujan's user avatar
22 votes
1 answer
886 views

Pseudo-holomorphic curves in the six-sphere

Equip $S^6$ with the almost complex structure coming from the cross product on $\mathbb R^7$ (i.e. the product on the pure imaginary octonions). What is known about the psudo-holomorphic curves in ...
John Pardon's user avatar
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66 votes
1 answer
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Is there an octonionic analog of the K3 surface, with implications for stable homotopy groups of spheres?

The infamous K3 surface has many constructions in many fields ranging from algebraic geometry to algebraic topology. Its many properties are well known. For this question I am really interested in the ...
Chris Schommer-Pries's user avatar
4 votes
1 answer
369 views

Action of G_2 on certain 7x7 skew-symmetric matrices

I have been working with something related to Goldman bracket for $G_2$ gauge group. There I have something like "$\text{Tr}(M_{\gamma}O_i)$", where $M_{\gamma}$ is a monodromy which takes value in ...
Hasib's user avatar
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1 vote
1 answer
135 views

Cayley Subspaces in a Calibrated 8-Space

Suppose we are given $(\mathbb{R}^8,\Phi)$, where $\Phi$ is the self-dual 4-form that defines $Spin(7)\subset SO(8)$ (Cayley calibration, see Notes on the Octonians, page 23). Now some 4-subspaces $V$ ...
Jjm's user avatar
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2 votes
1 answer
321 views

$Spin(7)$ as stabilizer of a $4$-form revisited

For a better understanding of this question, please see the question and answer here. In $Spin(8)$ there are plenty of copies of $Spin(7)$; consider, for instance, the antiimage of $SO(7)<SO(8)$ ...
Jjm's user avatar
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3 votes
1 answer
312 views

Octonions product: inversion in the right and identity in the left

Once octonions product is studied, together with the relations with $Spin(8)$ and $SO(8)$ geometry (see for instance Robert Bryant's notes), one realises that the key fact bringing all the phenomena ...
Jjm's user avatar
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8 votes
1 answer
392 views

$Spin(7)$ as stabilizer of a $4$-form

According to Bryant's work on special holonomy groups, $G_2\subset SO(7)$ may be defined as the group preserving the following 3-form: $\phi_0=\mathrm{d}x_{123}+\mathrm{d}x_{145}+\mathrm{d}x_{167}+\...
Jjm's user avatar
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5 votes
2 answers
409 views

How can the Cayley table for the elements of basis of a Cayley-Dickson algebra be summarized in an algebraic expression?

One would be able to construct a Cayley table that has all $e_i$ elements of the basis of algebra $A$ where $0<i<\dim A$ such that $e_0=1$, $e_1=i$, $e_2=j$ and so on. I'm looking for an ...
user272651's user avatar
0 votes
2 answers
629 views

Decomposition of $S^7=\operatorname{Spin}(7)/G_2$

$\DeclareMathOperator\Spin{Spin}$The seven-sphere can be written as the reductive space $S^7=\Spin(7)/G_2$. Has the decomposition $\Spin(7)=G_2\times S^7$ been calculated somewhere; maybe in terms of ...
Oliver Jones's user avatar
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6 votes
0 answers
155 views

Grunsky-Motzkin-Schoenberg formula

I found this formula in Brian McCartin's interesting book "Mysteries of the equilateral triangle" http://www.kettering.edu/news/mysteries-equilateral-triangle and it looks as follows: Suppose that $...
Zurab Silagadze's user avatar
22 votes
1 answer
2k views

Octonions and the Fano plane.

Does the Fano plane mnemonic for octonion multiplication have any deeper meaning? http://upload.wikimedia.org/wikipedia/commons/2/2d/FanoPlane.svg The symmetry group of the Fano plane is PSL(2,7), ...
Drew Armstrong's user avatar
5 votes
1 answer
376 views

Filling in a rational orthogonal matrix given one row

Quick version: given natural $n$ and a row of $n$ integers such that the sum of the squares is another square, call it $m^2.$ For $n=5,6,7$ is it always possible to fill in the rest of an $n$ by $n$ ...
Will Jagy's user avatar
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