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864 views

Quaternion between two quaternions [closed]

Hello, I have an orientation P1 in a 3D space, represented as a quaternion [w x y z]. Then P1 is rotated using another quaternion (q1) with the formula ...
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1answer
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In the quaternions, “any imaginary unit may be called i”

Introduction Suppose we are trying to prove that $\rm PSO_3\times PSO_3$ is isomorphic with $\rm PSO_4,$ and we catch on to the idea of using the quaternions to do so. We realize (as in Conway & ...
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5answers
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Is there a quaternionic algebraic geometry ?

Let $\mathbb{H}$ be the skew-field of quaternions. I'm aware of the Theorem 1. A function $f:\mathbb{H}\to\mathbb{H}$ which is $\mathbb{H}$-differentiable on the left (i.e. the usual limit ...
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William Rowan Hamilton and Algebra as Time

This question ended up longer than I intended (though most of the bulk is interesting remarks by Hamilton), so I thought it might be good to include my question at the beginning before the ...
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Why is every quadratic subfield of a Galois extension of the rationals with the quaternions as Galois group real?

Suppose that L is a field extension of the rationals with Galois group the quaternions Q={1,-1,i,-i,j,-j,k,-k}. Furthermore assume that L contains a quadratic subfield K. I have learned from this link ...
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Gaussian primes, quaternion primes, … octonions?

Is there a notion of an octonion prime? A Gaussian integer $a + bi$ with $a$ and $b$ nonzero is prime if $a^2 + b^2$ is prime. A quaternion $a + bi + cj + dk$ is prime iff $a^2 + b^2 +c^2 + d^2$ is ...
10
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1answer
424 views

What rings/groups have interesting quaternionic representations?

Let $\mathbb{H}$ denote the quaternions. Let $G$ be a group, and define a representation of $G$ on $\mathbb{H}^n$ in the natural way; that is, its a map $\rho:G\rightarrow ...