The tag has no usage guidance.

learn more… | top users | synonyms

11
votes
1answer
243 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 ...
47
votes
1answer
860 views

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 ...
4
votes
1answer
209 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 ...
1
vote
1answer
81 views

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 $V$ ...
1
vote
1answer
138 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 $SO(7)<SO(8)$ ...
2
votes
0answers
145 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 ...
7
votes
1answer
207 views

$Spin(7)$ 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: ...
2
votes
1answer
81 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 ...
-1
votes
1answer
261 views

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?
4
votes
0answers
90 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 ...
16
votes
1answer
739 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), ...
5
votes
1answer
232 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 $n$ ...
16
votes
1answer
818 views

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 ...
6
votes
3answers
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 ...