Questions tagged [quaternions]
The quaternions tag has no usage guidance.
136
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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 ...
46
votes
6
answers
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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 $h^{-1}\...
32
votes
1
answer
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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 ...
27
votes
1
answer
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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
...
26
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5
answers
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Is it possible to "get" quaternions without specifically postulating them?
I know that quaternions were first invented to handle description of 3D and 4D rotations, just as 2D rotations can be described by complex numbers. On the other hand, non-natural numbers can be "...
26
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2
answers
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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 admittedly-...
25
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1
answer
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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 ...
22
votes
1
answer
575
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A symmetric-like group and the quaternion group $Q_8$
It is well known that the symmetric group $S_n$ admits presentation with
$\{(ij) \mid i\neq j\}$ as the set of generators and the following list of relations
(in every formula distinct letters denote ...
17
votes
2
answers
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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 ...
17
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1
answer
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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{...
15
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Why is the automorphism group of the octonions $G_2$ instead of $SO(7)$
I can calculate the derivation of the octonions and I clearly find the 14 generators that form the algebra of $G_2$. However, when I do the same calculation for the quaternions, I end up with the ...
15
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1
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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,...
12
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0
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Is there a mistake in Zagier's "Hyperbolic manifolds and special values of Dedekind zeta-functions"?
I am having troubles with Don Zagier's "Hyperbolic manifolds and special values of Dedekind zeta-functions", available at this link, and I think there might be a mistake. In particular the ...
12
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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 ...
11
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2
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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 ...
11
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1
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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.
...
10
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2
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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 ...
10
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2
answers
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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
votes
1
answer
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Diagonalizing quaternionic unitary matrices
The quaternionic unitary group $\mathrm{U}(n,\mathbb{H})$, also called the compact symplectic group $\mathrm{Sp}(n)$, consists of $n \times n$ quaternionic matrices $g$ such that $gg^\ast = 1$, where
$...
10
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1
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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 ...
10
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0
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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 ...
10
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0
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417
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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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2
answers
677
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Maximal compact subgroup of $\mathrm{SL}(2,\mathbb{H})$
$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\SL{SL}$The exceptional isomorphism $\Spin(5,1)\simeq \SL(2,\mathbb{H})$ is well-known, and I can find references that ...
9
votes
1
answer
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Representing a number as a sum of four squares and factorization
Rabin and Shallit have a randomized polynomial-time algorithm to express an integer $n$ as a sum of four squares $n=a^2+b^2+c^2+d^2$ (in time $\log(n)^2$ assuming the Extended Riemann Hypothesis).
I'...
9
votes
1
answer
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Existence of certain endomorphism of supersingular elliptic curve
Let $E$ be a supersingular elliptic curve over an algebraically closed field $K$ of characteristic $p$. Let $R = \operatorname{End}(E)$ be its ring of endomorphisms. Then, it is known $R \otimes_{\...
8
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2
answers
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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 $Q$...
8
votes
1
answer
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Covolumes of unit groups of division algebras
Let $D$ be a central division (or maybe just simple) algebra over $\mathbb{Q}$. Let $\mathcal{O} \subset \mathcal{O}_m$ be an order inside a fixed maximal order and denote by $\mathcal{O}^1$ its group ...
8
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2
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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
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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 ...
8
votes
2
answers
663
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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}(...
8
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3
answers
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What other lattices are obtainable from this noncommutative ring?
Here I will regard $SU(2)$ as the multiplicative group of unit quaternions.
There are just three conjugacy classes of finite subgroups $G < SU(2)$ where $[G:C] > 2$ for all cyclic subgroups $C &...
7
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2
answers
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Automorphisms and isometries of the quaternions
Each automorphism of the quaternion algebra is inner and it is an orthogonal mapping with the determinant 1.
Let $f: \mathbb H \rightarrow \mathbb H$ be in $SO(4)$. Does there exists a quaternion $q$...
7
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3
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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 ...
7
votes
1
answer
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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
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5
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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 ...
6
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1
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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 & ...
6
votes
4
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Generalizing contour integration to quaternions and bicomplex numbers
I am interested in the possibility of generalizing the notion of contour integration to the quaternions or bicomplex numbers. I am aware that the Frobenius theorem prevents the construction of a true ...
6
votes
2
answers
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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 ...
6
votes
2
answers
465
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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}\...
6
votes
4
answers
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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 $UP$, ...
6
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2
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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 ...
6
votes
1
answer
572
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The Hilbert symbols of quaternion algebras over a totally real field
Let $k$ be a totally real number field. Every quaternion algebra $B$ over $k$ can be written as
$$B = \left(\frac{a,b}{k}\right), $$
for some constants $a,b \in k^\times$. My question is, can I always ...
6
votes
1
answer
290
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Idempotent functions on Sp(1)
The quaternion group $Sp(1)\simeq S^3$ can be understood as $(z,w)\in\mathbb {C}^2$ with $|z|^2+|w|^2=1$ where multiplication is defined by $(z,w)(t,s)=(zt-\bar{s}w,zs+\bar{t}w)$.
Question: How do ...
6
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1
answer
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Computing Tamagawa number of torus in Quaternion algebra
Consider a rational Quaternion algebra $M$ over $\mathbb{Q}$ that does not split at $\infty$. For example take the rational Hamilton quaternions $M=\mathbb{Q}(-1,-1)$.
For the adele ring $\mathbb{A}$ ...
6
votes
1
answer
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Conductor of quaternionic representation
Classical work by Casselman shows that for an irreducible admissible representation $\rho$ of $GL_2$ over a non-archimedean field $k$, there is a minimal power $n\geq 0$ of the prime ideal $\mathfrak{...
6
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0
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Is there a residue sum formula in quaternionic analysis?
In complex analysis, there is a formula involving the residues of complex functions that one can employ to find the value of certain infinite series.
If the function $f: \mathbb{C} \to \mathbb{C} $ ...
6
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0
answers
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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 $...
5
votes
4
answers
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Solutions to the Diophantine equation $x^2+3y^2+3z^2=n$
For a fixed positive integer $n$, the Diophantine equation
$$x^2 + y^2 + z^2 = n$$
was studied by Gauss in Disquisitiones Arithmeticae. As is known, this equation is intimately connected to the ...
5
votes
2
answers
291
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Automorphic quotient for quaternion algebras
Are automorphic quotient for quaternion algebras always compact (safe the totally split case)?
Is there any good reference for proof of this fact, or easy arguments to say do?
5
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2
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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 "...