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Results tagged with octonions
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user 13972
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.
15
votes
Accepted
Why, conceptually, does the torus normalizer in $G_2$ split?
Here's a description that doesn't use octonions; instead, it uses the definition of $\mathrm{G}_2$ as the stabilizer of a $3$-form on $\mathbb{R}^7$. …
- 108k
7
votes
Accepted
Quadratic forms on $\mathbb{R}^{16}$ coming from octonions
-x\,I_8 }\
:\
\matrix{ x\in\mathbb{R},\cr {\bf w},{\bf x}\in\mathbb{O},\cr
\ a\in{\frak{spin}}(8)}\ \right\}
\subset{\frak{sl}}(16,\mathbb{R})\,,
$$
where $C$ denotes conjugations in the octonions … , $R_{\bf w}$ denotes right multiplication by ${\bf w}$ in the octonions, $L_{\bf x}$ denotes left multiplication by ${\bf x}$ in the octonions, and the assignments $a\mapsto a_k$ for $k=1,3$ are Lie algebra …
- 108k
5
votes
Diagonalization of octonionic Hermitian matrices of size $2\times 2$
Yes, in fact, any $2$-by-$2$ octonionic Hermitian matrix is equivalent under the natural $\mathrm{Spin}(9)$ action to a diagonal $2$-by-$2$ octonionic Hermitian matrix.
This follows from the well-know …
- 108k
27
votes
Accepted
Why is the automorphism group of the octonions $G_2$ instead of $SO(7)$
Hence the automorphism group of the octonions is a Lie group of dimension $14$.
Note that, any automorphism of the octonions that fixes three such elements is the identity. … imaginary octonions. …
- 108k
10
votes
Accepted
Characteristic polynomial of an $8 \times 8$ symmetric matrix with indeterminate entries rel...
Let $a,b,c\in\mathbb{O}$ be octonions and consider the linear map $L:\mathbb{O}\to\mathbb{O}$ defined by
$$
L(x) = (b(cx))a = R_aL_bL_c(x).
$$
One desires a formula for the characteristic polynomial of …
- 108k
29
votes
Accepted
Pseudo-holomorphic curves in the six-sphere
It might be immodest of me to mention my own work, but there is quite a lot known about the pseudo-holomorphic curves in $S^6$. For example, it is known that the pseudoholomorphic rational curves are …
- 108k
5
votes
Accepted
Properties of complexified octonions
The complexified octonions satisfy $N(xy) = N(x)N(y)$, and the norm is a nondegenerate quadratic form on $\mathbb{C}{\otimes}\mathbb{O}$, considered as a complex vector space.
1) No. …
- 108k
3
votes
Accepted
Cayley Subspaces in a Calibrated 8-Space
All your questions (and much more) are answered in the 1982 paper Calibrated Geometries by Harvey and Lawson. See Acta Mathematica July 1982, Volume 148, Issue 1, pp. 47-157.
Just so you'll know: H …
- 108k
5
votes
Accepted
$Spin(7)$ as stabilizer of a $4$-form revisited
I think that you want to look at the relevant passages in Spin Geometry by Lawson and Michelsohn, particularly Chapter IV, Sections 9 and 10, where they explain in general how the square of a spinor c …
- 108k
11
votes
Accepted
$Spin(7)$ as stabilizer of a $4$-form
Harvey and Lawson, Calibrated geometries (Acta Math. 1982) proves this (i.e., that the stabilizer of $\Omega_0$ is isomorphic to the nontrivial double cover of $\mathrm{SO}(7)$) using properties of the octonions … Second you can find a proof that doesn't rely on knowledge of the octonions but instead relies on the classification of Lie algebras and groups in Bryant's Metrics with exceptional holonomy (Ann. of Math …
- 108k