Questions tagged [quaternions]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
6 votes
0 answers
253 views

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} $ ...
user avatar
  • 3,463
5 votes
1 answer
136 views

Why is $\operatorname{U}(n,\mathbb{H})\subset \operatorname{SL}(n,\mathbb{H}) $?

This question is inspired by Maximal compact subgroup of $\mathrm{SL}(2,\mathbb{H})$. Consider the embedding $\operatorname{U}(n,\mathbb{H})\subset \operatorname{GL}(n,\mathbb{H}) $. Since $\...
user avatar
  • 34.2k
0 votes
1 answer
58 views

Maximizing a skew-symmetric 4D cross product

How do I find two orthonormal 4D vectors, $(x_0,x_1,x_2,x_3)$ and $(y_0,y_1,y_2,y_3)$, which maximize this function: $-19x_1y_0 - 33x_2y_0 + 11x_3y_0 + 19x_0y_1 - 21x_2y_1 - 33x_3y_1 + 33x_0y_2 + ...
user avatar
  • 1,425
3 votes
0 answers
63 views

Nonassociative quaternion algebras

I'm interested in nonassociative central simple algebras; I've found Lee and Waterhouse's article 'Maximal Orders in Nonassociative Quaternion Algebras', and this cites a previous article by ...
user avatar
  • 221
8 votes
1 answer
217 views

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 ...
user avatar
  • 697
1 vote
0 answers
102 views

Conway's quaternion notation +1/3[C_3×C_3]∙2^(2) represent C_3h of 4d Point group?

In John Conway and Derek Smith's On Quaternions and Octonions: their Geometry, Arithmetic, and Symmetry, they introduce a way to connect quaternions to 4D Point Group. Suppose: $[l,r]:x\to \bar lxr\;,\...
user avatar
  • 11
1 vote
0 answers
87 views

Sequences generated from commuted quaternions and general commuted linear transformations

Given a pair of non-commuting linear transformations, $A$ and $B$, define the "next pair" in a sequence as $A*B$ and $B*A$. I am interested in finite cycles (i.e., the sequence eventually ...
user avatar
  • 1,425
3 votes
3 answers
369 views

Ramification of quaternion algebras over $\mathbb Q$

I'm reading on the classification of quaternion algebras over $\mathbb{Q}$. In the most common definition, we have a quaternion algebra $Q = \left(\frac{a,b}{\mathbb{Q}} \right)$ splits at a finite ...
user avatar
  • 791
2 votes
0 answers
62 views

Bézout theorem for complex solutions of systems of homogeneous quaternionic polynomials

Suppose that I have a system of $n$ homogeneous polynomials in $n+1$ variables $p_{1},\dots,p_{n}\in\mathbb{H}[x_{0},\dots,x_{n}]$ with coefficients at both sides in $\mathbb{H}.$ I want to know if ...
user avatar
  • 522
14 votes
1 answer
418 views

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 ...
user avatar
  • 305
6 votes
4 answers
387 views

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 ...
user avatar
  • 475
3 votes
1 answer
102 views

Charaterisation of quaternion algebras

Let $k$ be a field, and $A$ an associative $k$-algebra with an identity element. Say that $A$ is quadratic if any subalgebra of $A$ generated by a single element has dimension at most two. I am ...
user avatar
  • 298
9 votes
1 answer
368 views

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 $...
user avatar
  • 19.3k
3 votes
1 answer
194 views

Quaternions as eigenvalues of rank 3 tensors

Let us consider a matrix $M^{(a)}$ of size $N \times N$, having $N$ eigenvalues $\lambda_i \in \mathbb{C}$. Considering a rank-3 tensor, we can informally think of it as a sequence of $N$ matrices $M^{...
user avatar
  • 125
1 vote
1 answer
179 views

Norm of maximal order in quaternion algebra

Let $B$ be a quaternion algebra over $\mathbb{Q}$ that is a division algebra over $\mathbb{R}$. The theorem of Hasse-Schilling tells us that the image of the reduced norm of $B^\times$ is $\mathbb{R}^+...
user avatar
  • 123
2 votes
0 answers
147 views

Two generalizations of the Verblunsky Theorem

I learned from this paper about the Verblunsky theorem. My question is that: What kind of generalizations of this theorem is availlable? In particular I am interested in the following two possible ...
user avatar
2 votes
0 answers
127 views

Sylow 2-subgroups of finite groups in which every subgroup of order 4 is cyclic

Let $G$ be a finite group. Assume that every subgroup of order 4 in $G$ is cyclic (as happens if $G$ is a cyclic group or a generalized quaternion group). It seems to me that it should follow that a ...
user avatar
1 vote
0 answers
85 views

Control and observability of Clifford algebra and quaternion valued systems?

Good evening everybody, I'm asking you if there is a mathematical theory about control and observability of differential systems within the quaternionic and Clifford (geometric) algebras. And if so, ...
user avatar
3 votes
1 answer
236 views

Question about the correspondence between unitary Möbius transformations and quaternions

One of the main theorems about the classification of Möbius transformations states that pure rotations of the Riemann sphere (without translation and dilatation) correspond to unitary Möbius ...
user avatar
  • 1,229
2 votes
1 answer
205 views

3D similarities and quaternions?

As is well-known, in dimension 2, a linear map $f : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ is a direct similarity if, once we identify $\mathbb{R}^2$ with $\mathbb{C}$, $f$ is of the form $$\forall z \...
user avatar
  • 2,010
1 vote
1 answer
248 views

Are there algorithms for deciding or solving conjugacy in integer quaternion rings?

I am doing some research on the quaternions and their role in Non-commutative cryptography. I have found a number of articles, but it is still unclear to me if there is a known solution to the ...
user avatar
0 votes
1 answer
234 views

Conjugacy in the quaternion group

Let $G$ be a non-commutative group, and suppose we are given two elements $x, y \in G$ which are conjugate, i.e. we know there exists some $z \in G$ such that $zxz^{-1} = y$. Can we find $z$ given $x$ ...
user avatar
  • 1,643
7 votes
3 answers
563 views

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 &...
user avatar
  • 3,425
4 votes
2 answers
551 views

Principled construction of the quaternions

Is there a construction of the quaternions that doesn't proceed through generators and relations, and which makes the connection with 3D rotations clear? I'm not happy with Clifford Algebra as an ...
user avatar
  • 3,321
8 votes
1 answer
641 views

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'...
user avatar
  • 61.2k
6 votes
1 answer
271 views

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 ...
user avatar
  • 1,533
3 votes
1 answer
213 views

How many distinct quaternions have a given prime norm $p$?

I seem to recall that the answer is $p + 1$, but I'm not quite sure.
user avatar
  • 1,643
3 votes
2 answers
418 views

The name of special 16-dimensional hypercomplex number

Let's consider the following number: $n = n_0 + n_1\textbf{i} + n_2\textbf{j} + n_3\textbf{k}$ Where $\{1,\textbf{i},\textbf{j},\textbf{k}\}\subset\mathbb{H}$ and in turn each component can be written ...
user avatar
3 votes
0 answers
71 views

Flag variety over quaternions and its Hecke algebra

Consider cell decomposition of flag variety of $\mathbb{H}^{n}$ into orbits under the action of $B(\mathbb{H})$ - group of upper triangular matrices with coefficients in $\mathbb{H}$. I think cells ...
user avatar
6 votes
1 answer
516 views

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 ...
user avatar
  • 517
3 votes
1 answer
244 views

Dimension of hermitian rank at most $k$ matrices over quaternions

In a (right) finite dimensional quaternionic Hilbert space there is an analogue of the spectral theorem (see theorem 4.6 in Farenick and Pidkowich) for normal matrices in $\mathbb{H}^{m\times m}$, ...
user avatar
  • 3,084
6 votes
1 answer
180 views

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{...
user avatar
  • 469
3 votes
2 answers
261 views

Quaternion holomorphic maps via certain elliptic operator instead of immediate generalization of complex differentiability

We identify $\mathbb{R}^4$ with the quaternions $\mathbb{H}=\{t=x+yi+zj+wk\mid x,y,z,w\in \mathbb{R}\}$. We define the differential operator $D$ on $C^{\infty}(\mathbb{R}^4)$, the space of smooth ...
user avatar
9 votes
1 answer
280 views

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_{\...
user avatar
1 vote
0 answers
108 views

Is it possible to prevent gimbal lock on a 2 axis gimble by adding a 3rd axis treating it as an "2d" quaternion? [closed]

If I have a 2 axis gimbal, that functions similar to a theodolite, that rotates the pitch and yaw axis. If I were to add an axis to make it a traditional gimbal, would I be able to prevent the gimbal-...
user avatar
5 votes
4 answers
585 views

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 ...
user avatar
  • 1,369
1 vote
1 answer
146 views

Name and Properties of Quaternions Related to 3D Differential Geometry

In 3D differential geometry there are canonical local coordinate systems associated with each point of sufficiently smooth curves and surfaces: in the case of curves there are the Frenet frames $$\...
user avatar
2 votes
0 answers
443 views

Symplectic group over finite field and quaternions

Symplectic group over finite field is defined as group preserving non-degenerate antisymmetric bilinear form on $\mathbb F_q^{2n}$. How could we define this group using quaternions ? This should be ...
user avatar
6 votes
1 answer
284 views

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 $\mathbb{A}$ ...
user avatar
5 votes
2 answers
255 views

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?
user avatar
  • 541
4 votes
1 answer
282 views

Why is this mapping surjective?

It is mentioned in wikipedia that every single orthogonal $3 \times 3$ rational matrix is of the form $$\dfrac{1}{m^2+n^2+p^2+q^2}\begin{pmatrix} m^2+n^2-p^2-q^2 & 2np-2mq & 2mp+2nq \\ 2mq+...
user avatar
4 votes
1 answer
396 views

Inner forms of $GL(2)$

I know that inner forms of $GL(2)$ are quaternion algebras. However, I cannot find the proof myself. First, since quaternion algebras are forms of $GL(2)$ by the usual embedding in matrices, they ...
user avatar
3 votes
1 answer
547 views

English translation of M.-F. Vigneras "Arithmétique des algèbres de quaternions"

I am looking to understand a citation about the connection of quaternion algebra over number fields which when embedded into $\mathbb{C}$, leads to a discrete subgroup of $SL_2(\mathbb{C})$ which ...
user avatar
1 vote
1 answer
190 views

Do these definitions of integrable quaternionic structure agree?

I have found two different definitions for integrable quaternionic structure in the literature, and I need to know if they agree with one another. One definition that I have found (from Differential ...
user avatar
  • 1,033
1 vote
0 answers
48 views

Which hyperkaehler manifolds admit an atlas with affine overlap maps?

In order for the quaternionic structure on a hyperkahler manifold to take the canonical form $$ J_1= \left[\begin{array}{cccc} 0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ ...
user avatar
  • 1,033
3 votes
1 answer
200 views

How does the concept of a hermitian metric generalize to a hyperkahler manifold?

A complex manifold admits an almost complex structure, $J$, which satisfies $$ {J^i}_j{J^j}_k=-{\delta^i}_k, $$ and a Hermitian metric, $g$, which satisfies $$ g_{st}{J^s}_i{J^t}_j=g_{ij}. \tag{1} $$...
user avatar
  • 1,033
3 votes
0 answers
166 views

Terminology: Almost hyper-Hermitian vs Almost hyper-Kähler vs

After non-thorough literature search, it seems to me that there is no consensus on the usage of the terminology "(almost) hyper-Hermitian" vs "almost hyper-Kähler" vs "almost quaternion-Hermitian" etc....
user avatar
  • 1,307
2 votes
2 answers
143 views

Matrix expression for elements of $\text{SO}_0(1,4)$

Denote by $\text{SO}_0(1,4)$ the identity component of the special linear isometry group $\mathrm{SO}(1,4)$ of the Lorentz-Minkowski space $\mathbb{R}_1^5$, that is, of $$\text{SO}(1,4)=\left\{X\in\...
user avatar
  • 207
4 votes
1 answer
868 views

Spherical Harmonics on $S^3$ [closed]

My understanding is that harmonic analysis on the circle ($S^1$) leads to Fourier Series/Integrals whereas harmonic analysis on the sphere ($S^2$) leads to Spherical Harmonics. If we take the next ...
user avatar
  • 157
7 votes
2 answers
2k views

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 $q$...
user avatar
  • 165