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5
votes
2answers
451 views

Book on ideal theory in Hurwitz quaternions

Hello, I am looking for a book that studies the set of Hurwitz quaternions (HQ). In particular, I am interested in a connection between HQ and imaginary quadratic fields (IQF); quaternion orders $\...
5
votes
2answers
716 views

Explicit description of a quaternion algebra with a prescribed set of ramified places

Let $k$ be an algebraic number field. I understand that given a finite set of non-complex places $S\subset V(k)$ of even cardinality, there exists a unique quaternion algebra $Q$ over $k$ such that $Q$...
1
vote
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 b_n$...
15
votes
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
votes
1answer
464 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 $P,...
3
votes
4answers
622 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
votes
0answers
381 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
votes
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
votes
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
votes
1answer
617 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
votes
3answers
617 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
votes
1answer
977 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 P2=q1*P1*q1'...
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
votes
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 $h^{-1}\...
21
votes
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 admittedly-...
5
votes
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 ...
11
votes
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
votes
1answer
458 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 Hom_{\mathbb{H}-}(\mathbb{...