As of May 31, 2023, we have updated our Code of Conduct.

# 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.

49 questions
Filter by
Sorted by
Tagged with
82 views

### 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, ...
44 views

### 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 ...
1 vote
67 views

### 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?
404 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 ...
413 views

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)...
106 views

### 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 ...
194 views

### 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 ...
112 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  about the existance of a projective plane $\left(\mathbb{C}\otimes\mathbb{O}\right)P^{2}$ defined on bi-...
616 views

### 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 ...
181 views

### 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 ...
536 views

### 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 ...
115 views

### 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 ...
184 views

### 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 ...
362 views

445 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,...
75 views

### 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. ...
248 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 ...
532 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?
470 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 ...
867 views

### 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 ...
359 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 ... 1 vote
167 views

396 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 ...
### 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 ...