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Questions tagged [octonions]

For questions on the octonions, a normed division algebra over the real numbers. It is a non-associative higher-dimensional analogue in the hierarchy of real, complex, and quaternionic numbers.

21 questions with no upvoted or accepted answers
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11
votes
0answers
221 views

Does the exceptional Lie algebra $\mathfrak{g}_2$ arise from the isometry group of any projective space?

I learned from Baez's notes on octonions that the classical simple Lie algebras can be identified with the Lie algebras of isometry groups of projective spaces over $\mathbb{R}, \mathbb{C}$ and $\...
8
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0answers
387 views

Fractal derivative of complex order and beyond

Is there some precise definition of "complex (fractal) order derivative" for all complex number? I am aware of the Riemann-Liouville fractional definition given here: Complex derivative but I would ...
6
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0answers
214 views

Generators of 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 isomorphic to $\mathfrak{so}(7)$. Can ...
6
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0answers
158 views

Are complex split-octonions isomorphic to a more easily-defined algebra?

I write fiction and nonfiction, both which are mathy. My fiction is not usually supermathy but I'm working on a fictional story that has some math in it, and I prefer accuracy to mathbabble. I'm ...
5
votes
1answer
265 views

What is a good book on general octonion algebras and the Cayley-Dickson construction?

I want a good reference that discusses properties of octonion algebras, especially over number fields. I'd like to know more about how this generalizes when applying the Cayley-Dickson construction ...
3
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0answers
80 views

A real example of an Octonion product

Goal: find the general Octonion multiplication product like the Quaternion formula given here: https://en.wikipedia.org/wiki/Quaternion#Multiplication_of_basis_elements ...
3
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0answers
47 views

Java Library that supports Quaternions Octonions, Sedenions?

I would like to experiment with multi dimensional complex numbers such as quaternions octonions, sedenions. I know Apache Commons Maths supports Quaternions, and I've found (although cannot download) ...
3
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0answers
47 views

The isotropy subgroup of the action of Spin(7) on the Grassmannian G(3,8)

The group Spin$\left( 7\right) \ $is the universal cover of $SO\left( 7\right) $ and is characterized as the subgroup of $SO\left( 8\right) $ consisting of the automorphisms of the triple cross ...
3
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0answers
61 views

The order of elements in finite octonions

After great success of this question I would like to continue this thread. Choose any element $x$ in algebra $\mathbb O_q$ of octonions over finite field $\mathbb F_q$. There is following equation ...
2
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0answers
71 views

Construction of the Octonions

I get how to construct number systems: $\Bbb{N,~Z,~Q,~R,~C}$. But what about the hypercomplex? How do we go from $\Bbb C$ to $\Bbb H$ and $\Bbb O$? I've also heard it stated that the octonions are ...
2
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0answers
29 views

Hermitian Octonion matrices?

The space of $n\times n$ octionion matrics $A$ such that $AA^*=A^*A=I_n$ is not a Lie group due to lack of associativity. But is it a smooth manifold? What is its dimension?
2
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0answers
22 views

If there can exist a model of the octonions without complex numbers

I know the ordering of complex numbers -> quaternions -> octonions, so the octonions are basically built on top of the complex numbers. But I am wondering, still, if there is a way to reformulate them ...
2
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1answer
139 views

Can one test an octonionic interpretation for a conjecture, apparently valid in the complex and quaternionic settings, and proven in the real case?

For the values $\alpha = \frac{1}{2},1, 2$, corresponding to real, complex and quaternionic scenarios, the formulas (https://arxiv.org/abs/1301.6617, eqs. (1)-(3)) \begin{equation} \label{Hou1} P_1(\...
2
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0answers
137 views

Motivating the Cayley-Dickson construction by proving Hurwitz's theorem

To me it seems the way to motivate the Cayley-Dickson construction is to prove Hurwitz's theorem, which is done over at Wikipedia. The theorem states the only real division algebras equipped with a ...
1
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0answers
18 views

Octonion Grassmanian?

I just learnt that the octonion projective space $\mathbf{OP}^n$ exists only for $n=1,2$. How about the "octonion Grassmannian", i.e. the space of $k$-subspace of $\mathbf O^n$? For which values of $(...
1
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0answers
200 views

Is octonion structure related to the fact that an 8-cube has an equal number of 2-faces and 3-cells?

Background: We know that octonions exist in 8-space, and we know that in 8-space, the 8-dimensional "measure polytope" ("hypercube") just so happens to have the SAME number of 2-dimensional faces (...
1
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0answers
184 views

How can we express determinant in terms of trace

The determinant of a matrix in $\mathfrak h_3(\Bbb O)$ is defined by $(a,b,c,\mathbf{a},\mathbf{b},\mathbf{c})=\begin{bmatrix} a &\mathbf{c} &\mathbf{b} \\ \mathbf{\bar{c}} & b & \...
1
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0answers
63 views

Octonions - affine space

I'm writing a project on Cayley's algebra. I have some topics which I have to follow and I've managed to solve most of them,except 2. I have written about their rule of multiplication,together with ...
1
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0answers
413 views

Has the Riemann Hypothesis been generalized to the Octonions and the Quaternions?

I've noticed that it uses imaginary numbers. I know that sometimes when I have too few dimensions like (-1)^n, dots show where I might expect lines due to imaginary numbers. So perhaps there is a ...
0
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0answers
12 views

Subalgebras of the Real Octonions under Multiplication

I am interested to know what the subalgebras of the octonions look like. I tried searching for it but to no avail. What did show up was "Subalgebras of the Split Octonions," which was quite nice. Yet, ...
0
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0answers
21 views

Commutation or Anti-commutation of the corresponding imaginary units of the octonions and split-octonions

Given a general octonion x: $\mathbb{O}$=$\mathbb{H}$+$\mathbb{H}$$L$ by x=$x^1$+$x^2$i+$x^3$j+$x^4$k+$x^5$i$L$+$x^6$j$L$+$x^7$k$L$+$x^8$$L$ with $L^2$=-1, and a general split-octonion x: $\...