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.
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Are H$_n$(O) (n>3) Jordan algebras?
As we know, H$_3$(O) is a 27-dimensional exceptional Jordan algebra, here O is Cayley octonion algebra.But how about n>3? I guess that when n>3, H$_n$(O) are not Jordan algebras. But I only have a ...
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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 ...
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Math beyond Quaternions
Quaternions remove the commutative property and octonions eliminate the associative property can we go any higher and eliminate more properties?
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What do we lose passing from the reals to the complex numbers?
As normed division algebras, when we go from the complex numbers to the quaternions, we lose commutativity. Moving on to the octonions, we lose associativity. Is there some analogous property that we ...
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A question about hypercomplex numbers: quaternions, octonions etc [duplicate]
Possible Duplicate:
Why is 8 so special?
First of all let me state that I am not a mathematician but I work in computer science and engineering.
I was reading about hypercomplex numbers, ...
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Constructing a parallelization of the 7-sphere.
I would like to show that $S^7$, the 7-sphere, is a parallelizable manifold. Let $\mathcal{O}$ be the octonions, the normed division algebra (noncommutative, nonassociative) over $H\times H$, where ...
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Why are properties lost in the the Cayley-Dickson construction?
Motivating question: What lies beyond the Sedenions?
I'm aware that one can construct a hierarchy of number systems via the Cayley-Dickson process:
$$\mathbb{R} \subset \mathbb{C} \subset ...
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Matrix Representation of Octonions
Since quaternions $\mathbb{H}$ have a matrix representation as elements of $\text{SU}(2,\mathbb{C})$ as the following
$$ 1 \mapsto \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix},\quad \mathrm ...
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Why is 8 so special?
I have been reading about multi-dimensional numbers, and found out that it's been proven that Octonions are the division algebra of the largest dimension. I was wondering why despite having infinitely ...
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Octonionic Hopf fibration and $\mathbb HP^3$
Real, complex and quaternionic versions of Hopf fibration ($S^0\to S^n\to\mathbb RP^n$, $S^1\to S^{2n+1}\to\mathbb CP^n$ and $S^3\to S^{4n+3}\to\mathbb HP^n$) give rise to spherical fibrations ...
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Nonzero Octonions as a 7-sphere
While reading about Moufang loops in the book "An introduction to Quasigroups and their Representations" by Smith, I've encountered the following statement:
The set $ S^7 $ of nonzero octonions of ...
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What are some real-world uses of Octonions?
... octonions are the crazy old uncle nobody lets out of the attic: they are nonassociative.
Comes from a a quote by John Baez. Clearly, the sucessor to quaterions from the Cayley-Dickson process ...
