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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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 ...
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9 votes
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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)...
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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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7 votes
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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 ...
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2 votes
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104 views

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-...
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14 votes
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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 ...
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5 votes
1 answer
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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 ...
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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 ...
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4 votes
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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 ...
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2 votes
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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 ...
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12 votes
1 answer
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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 = ...
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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 ...
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1 vote
2 answers
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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 ...
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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 ...
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17 votes
0 answers
642 views

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 (...
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22 votes
1 answer
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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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4 votes
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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 ...
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2 answers
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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,...
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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. ...
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2 votes
1 answer
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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 ...
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5 votes
2 answers
492 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?
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2 answers
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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 ...
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10 votes
1 answer
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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 ...
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6 votes
1 answer
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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
0 answers
155 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 $\...
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11 votes
1 answer
622 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. ...
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6 votes
0 answers
227 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 ...
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3 votes
1 answer
119 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 ...
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8 votes
1 answer
334 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 $a$, $b$, $c$...
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6 votes
1 answer
193 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-...
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5 votes
1 answer
279 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}\}, ...
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22 votes
1 answer
745 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 ...
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64 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 ...
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4 votes
1 answer
339 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 ...
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1 vote
1 answer
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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 $V$ ...
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2 votes
1 answer
280 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 $SO(7)<SO(8)$ ...
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3 votes
1 answer
300 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 ...
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8 votes
1 answer
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$Spin(7)$ 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}+\...
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5 votes
2 answers
378 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 ...
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0 votes
2 answers
570 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 ...
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6 votes
0 answers
140 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 $...
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20 votes
1 answer
1k 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), ...
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5 votes
1 answer
301 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 $n$ ...
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23 votes
1 answer
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The octonions on a bad day

We can define the algebra of quaternions $\mathbb H$ over any field $k$, and depending on the arithmetic of $k$ it is either a division algebra or a matrix algebra. We can also define the algebra of ...
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7 votes
3 answers
1k views

Realizing proper pure octonions as conjugates

Let us take the octonions as having all integer coefficients and the multiplication table at BAEZ We have a standard conjugation operator with $\bar{1} = 1$ and $\bar{e_i}= - e_i,$ extend by ...
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46 votes
11 answers
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What is the Cayley projective plane?

One can build a projective plane from $\Bbb R^n$, $\Bbb C^n$ and $\Bbb H^n$ and is then tempted to do the same for octonions. This leads to the construction of a projective plane known as $\Bbb OP^2$, ...
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