# 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.

37 questions
Filter by
Sorted by
Tagged with
97 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 ...
32 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 ...
313 views

425 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,...
71 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. ...
195 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 ...
421 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?
380 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 ...
773 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 ...
295 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 ...
146 views

296 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 $S^7=Spin(7)/G_2$
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 Cayley numbers?