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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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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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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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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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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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 ...
user21230
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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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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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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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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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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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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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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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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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$Spin(7)$ 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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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 $a$, $b$, $c$...
user21230
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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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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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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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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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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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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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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 ...
user21230
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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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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 ...
user21230
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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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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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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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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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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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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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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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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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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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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 ...
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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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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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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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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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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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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)...
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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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$Spin(7)$ 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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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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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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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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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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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?
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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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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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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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