The quaternions tag has no usage guidance.

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### Do unit quaternions at vertices of a regular 4-simplex, one being 1, generate a free group?

Choose unit quaternions $q_0, q_1, q_2, q_3, q_4$ that form the vertices of a regular 4-simplex in the quaternions. Assume $q_0 = 1$. Let the other four generate a group via quaternion ...

**36**

votes

**5**answers

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

**1**answer

2k 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-...

**20**

votes

**1**answer

1k views

### history of quaternion algebras

Who is responsible for the generalization of Hamilton's quaternions to other types of quaternion algebras, and when did this occur? In particular, Hamilton's quaternions are the 4-dimensional algebra ...

**19**

votes

**1**answer

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 ...

**15**

votes

**1**answer

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

**1**answer

467 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,...

**11**

votes

**2**answers

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 ...

**11**

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**0**answers

369 views

### Nullstellensatz for quaternionic plane curves?

By a quaternionic plane curve I mean the zero locus of a noncommutative polynomial in two variables, $x$ and $y$ say, over ${\Bbb H}$, Hamilton's quaternions. It is evidently well-known that, after ...

**10**

votes

**2**answers

368 views

### Which finite nonabelian groups have all their quaternionic representations of degree one?

A finite group $G$ has a finite set of irreducible representations over the complex numbers. All of these representations are linear (that is, are maps in 1x1 complex matrices) if and only if $G$ is ...

**10**

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**1**answer

460 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{...

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527 views

### Quaternions: ellipse effect

I would be interested in an explanation of the "six ellipse effect" produced by the pseudocode below (I also wonder how close are these to being actual ellipses). Note the code is somewhat similar to ...

**9**

votes

**2**answers

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 ...

**9**

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**1**answer

513 views

### Explicit isomorphism for quaternion algebras over $\mathbb{Q}$?

It is known that the isomorphism class of a quaternion algebra $A=\binom{a,b}{K}$ over a number field $K$ is determined by the finite set of places $v$ of $K$ where $A\otimes_K K_v$ is a division ...

**9**

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**0**answers

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 ...

**8**

votes

**2**answers

606 views

### Which quaternion algebras have class number one?

Over Q, the definite quaternion algebras with a unique conjugacy class of maximal orders, i.e. "with class number one", are those with discriminant 2,3,5,7, and 13.
Three questions:
What is a ...

**8**

votes

**2**answers

278 views

### Circle Action on Quaternionic Projective Space

Quoting from Wikipedia article on quaternionic projective space:
Therefore the quotient manifold
$$
\mathbb{HP}^{2}/\mathrm{U}(1)
$$
may be taken, writing $U(1)$ for the circle group. It has ...

**8**

votes

**1**answer

161 views

### Number of orbits of $\mathrm{SL}_2(\mathcal O_A)$ on $\mathbf P^1(A)$ when $A$ is a quaternion algebra

This is a reference request.
Let $A$ be an anisotropic quaternion algebra over $\mathbf Q$. Let $\mathcal O_A$ be a maximal order in $A$. Then $\mathrm{SL}_2(A)$ acts transitively on the right on ...

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votes

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136 views

### Determinants of octonionic hermitian matrices

For quaternionic hermitian matrices (i.e. quaternionic square matrices $(a_{ij})$ satisfying
$a_{ji}=\bar a_{ij}$) there is a nice notion of (Moore) determinant which can be defined as follows.
...

**7**

votes

**1**answer

160 views

### riemann mapping theorem for skew-fields of quaternions and beyond

Is there any known generalization of the riemann mapping theorem over skew-fields of quaternions and beyond or at least a conjectured formulation of it?
In a less focused way, how far does the main ...

**6**

votes

**2**answers

736 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 $Q$...

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582 views

### Hurwitz integers represented as sums of two squares of Hurwitz integers

I wonder if there exists a characterisation of Hurwitz integers which are represented as sums of two squares of Hurwitz integers, up to multiplication by a unit. And if so, could you please point to a ...

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### Realizing proper pure octonions as conjugates

Let us take the octonions as having all integer coefficients and the multiplication table at BAEZ
We have a standard conjugation operator with $\bar{1} = 1$ and $\bar{e_i}= - e_i,$ extend by ...

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votes

**2**answers

479 views

### Reference on ideal theory in Hurwitz quaternions

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 $\mathcal{O}(...

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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 ...

**5**

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332 views

### Volumes of fundamental domains of maximal orders in definite quaternion algebras over Q

I'm looking for an explanation of the following result:
If D is a maximal order in a definite (i.e. ramified at infinity) quaternion algebra B over $\mathbb{Q}$, and $\phi : B\otimes\mathbb{R}\...

**5**

votes

**2**answers

607 views

### How do you find maximal orders in quaternion algebras?

Let A be the 4-dimensional algebra over Q with basis 1,i,j,k, and multiplication table
$$
i^2 = -1 \quad j^2 = -11 \quad k^2 = -11 \quad ij = k \quad jk = 11i \quad ki = j
$$
So, A is the unique "...

**5**

votes

**1**answer

240 views

### Filling in a rational orthogonal matrix given one row

Quick version: given natural $n$ and a row of $n$ integers such that the sum of the squares is another square, call it $m^2.$ For $n=5,6,7$ is it always possible to fill in the rest of an $n$ by $n$ ...

**4**

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180 views

### Is an associative division algebra required for this phenomenon?

For which integers $d \geq 1$ can we find real matrices $R_1, \dotsc, R_d$ of size $d \times d$ such that for any unit vector $v \in \mathbb{R}^d$, $$R_1 v, \dotsc, R_d v$$ is an orthonormal basis? ...

**4**

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**1**answer

185 views

### Finding Finite Generators of a Subset of a Quaternion Algebra/Cocompact Lattices

I was wondering if anyone had some ideas (books, papers, experience) on how to explicitly compute generators for the elements of a quaternion algebra, $Q$, with reduced norm $1$. I'm trying to ...

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**1**answer

85 views

### Left- and right-sided principal ideals of quaternions have same index?

One fact about the Lipschitz integers (quaternions of the form $a + bi + cj + dk$ where $a, b, c, d$ are integers) is that the left-sided ideal generated by any element $Q$ has the same index in the ...

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162 views

### Hochschild cohomology of SU(2)

I have a question about the computation of an Hochschild Cohomology. Or at least about a space which really looks like a cohomology space. All the functions i consider are assumed to be smooth.
Let's ...

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102 views

### Grunsky-Motzkin-Schoenberg formula

I found this formula in Brian McCartin's interesting book "Mysteries of the equilateral triangle" http://www.kettering.edu/news/mysteries-equilateral-triangle and it looks as follows:
Suppose that $...

**3**

votes

**4**answers

628 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 $UP$, ...

**3**

votes

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620 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 ...

**3**

votes

**2**answers

246 views

### 3D Rotation Representation for Multiple Turns

Is there a rotation representation that can also represent "turns", instead of collapsing coincident rotations into the same representation?
In 2D, a simple angle satisfies this, as it can have ...

**3**

votes

**1**answer

432 views

### What does the Jacquet-Langlands correspondence say about quaternion algebras of class number one?

If F is a totally real number field of degree n, and A is a definite quaternion algebra over F, I understand (not really) the Jacquet Langlands correspondence to construct a modular form in n ...

**3**

votes

**1**answer

211 views

### The de Rham complex of a quaternion-Kahler manifold

As we all know, for a complex manifold $M$, its de Rham complex admits a decomposition into a double complex called the Dolbeault complex. If $M$ also admits a Kahler metric, then we get the wonderful ...

**3**

votes

**2**answers

736 views

### Skew fields inside quaternion division algebras

Suppose that $Q$ is a quaternion division algebra with center $k$, where $k$ is an arbitrary commutative field (let's say with $\operatorname{char}(k) \neq 2$ if necessary). Assume that $D$ is an ...

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**1**answer

58 views

### Quaternion orders such that every proper ideal is invertible

Let $B$ be a quaternion algebra over $\mathbb{Q}$ and let $\mathcal{O} \subset B$ be an order.
A lattice in $B$ is (left) proper over $\mathcal{O}$ if its left order is equal to $\mathcal{O}$. We ...

**3**

votes

**1**answer

238 views

### Question about the definition of the genus 0 curves in Gross' paper “Heights and the Special values of L-series”

Let $N \in \mathbb{Z}$ be a prime number, and let $B = \left( \dfrac{a, b}{\mathbb{Q}} \right)$ be the unique quaternion algebra over $\mathbb{Q}$ ramified at $N$ and at $\infty$. Then, in section 3 ...

**3**

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**1**answer

168 views

### On conductors, levels and traces on quaternion algebras

I am currently working on level issues in the division central simple algebra case, say $D$ over a local non-archimedean field $F$ (e.g. $\mathbf{Q}_p$). Let say that $\mathcal{O}_D$ and $\mathcal{O}...

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111 views

### Maximal compact subgroup of SL(2,|H)

The exceptional isomorphism $Spin(5,1)\simeq SL(2,\mathbb{H})$ is well-known, and I can find references that say the maximal compact of $Spin(5,1)$ is $Spin(5) \simeq Sp(2)$. So I know the answer to ...

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239 views

### Is the “algebraic closure” of the quaternions, finite dimensional? [closed]

This post is a sequel of: What's the algebraic closure of the quaternions?
$\mathbb{H}$ is algebraically closed for the polynomials of the form $\sum a_r x^r$, but it is not for the polynomials ...

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**1**answer

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 & ...

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### How can the Cayley table for the elements of basis of a Cayley-Dickson algebra be summarized in an algebraic expression?

One would be able to construct a Cayley table that has all $e_i$ elements of the basis of algebra $A$ where $0<i<\dim A$ such that $e_0=1$, $e_1=i$, $e_2=j$ and so on. I'm looking for an ...

**2**

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**2**answers

606 views

### ramified quaternion algebras

I'm trying to better understand the connection between the concepts of ramification of a field extension, and ramification of a quaternion algebra. I'm also trying to build a better understanding of ...

**2**

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**1**answer

619 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 $exp$ ...

**2**

votes

**1**answer

147 views

### Why do noncocompact arithmetic Kleinian groups have quadratic trace fields?

I realize there are a few different ways of going about proving this, depending on one's background, but there's a particular number theoretic aspect that I am just blanking on, and can't seem to find ...

**2**

votes

**1**answer

389 views

### Distance measure for noisy $SE(3)$ transforms

I have a transformation $T \in SE(3)$ parameterized by a mean quaternion $q$ with covariance matrix $\Sigma_q \in R^{4\times4}$ and a mean translation $t \in R^3$ with covariance matrix $\Sigma_t \in ...