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1
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1answer
191 views

regular quaternion functions of form f(q) =c + a*q*b + …

Hello, is it possible do somehow define regular function(in the way that some analogy to Cauchy integral formula would hold) over quaternions that function of form $$f(q)= \sum_{n=0}^\inf a_n q^n ...
15
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1answer
5k views

The gimbal lock shows up in my quaternions

I suspect this is a bit basic for mathoverflow, seeing I'm still just an undergraduate I've been playing around with quaternions as means to eliminate the gimbal lock. From what I understand, one ...
14
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1answer
459 views

Is the ring of quaternionic polynomials factorial?

Denote by $\mathbb{H}[x_1,\dots,x_n]$ the ring of polynomials in $n$ variables with quaternionic coefficients, where the variables commute with each other and with the coefficients. Two polynomials ...
3
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4answers
620 views

Polar decomposition for quaternionic matrices?

A non-zero complex number can be uniquely written in polar form as $re^{i\theta}$. There is an analogous result for complex matrices: any invertible complex matrix can be uniquely written as $UP$, ...
9
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0answers
380 views

Polynomial function from $S^3$ to $S^3$ and quaternions

I am searching the polynomial functions from $S^3$ to $S^3$. ($S^3$ is the set of vectors $x$ in $\mathbb{R}^4$ such that $\|x\|=1$) We say $g$ is a polynomial function from $S^3$ to $S^3$, if there ...
9
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2answers
1k views

When the determinant of a 2x2 polynomial matrix is a square?

Consider a 2x2 matrix $A$ with entries from $\mathbb{C}[x,y]$. Assume that $\mathrm{det} A$ is a square. Is it true that then $A$ can be represented as a noncomuting product $A=A_1 A_2 … A_{2n}$, in ...
19
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1answer
1k views

The Quaternion Moat Problem

"One cannot walk to infinity on the real line if one uses steps of bounded length and steps on the prime numbers. This is simply a restatement of the classic result that there are arbitrarily large ...
2
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1answer
615 views

Central division and quaternion algebras

I would like to know if there are some central-simple algebras $D_1$, $D_2$ and $D_3$ over a field $k$ satisfying the following properties : $ind(D_1)=exp(D_1)=4$ ($ind$ is the Schur index and $exp$ ...
3
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3answers
615 views

Polynomial Vector Fields on the 3-Sphere

EDIT(3): I am looking for a basis for the Lie algebra of polynomial vector fields on $S^3$. EDIT(2): I am fairly certain now that my question is more along the lines of, what does the Lie algebra of ...
0
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1answer
963 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 ...
2
votes
1answer
2k views

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 & ...
36
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5answers
2k views

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 ...
21
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1answer
1k views

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 ...
5
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5answers
2k views

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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2answers
2k views

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
456 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 ...