Questions tagged [quaternions]

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13
votes
1answer
331 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 ...
5
votes
4answers
264 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 ...
3
votes
1answer
89 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 ...
9
votes
1answer
308 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 $...
3
votes
1answer
134 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^{...
1
vote
1answer
80 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}^+...
2
votes
0answers
145 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 ...
2
votes
0answers
102 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 ...
1
vote
0answers
76 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, ...
3
votes
1answer
147 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 ...
2
votes
1answer
154 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 \...
1
vote
1answer
235 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 ...
0
votes
1answer
183 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 $x$ ...
7
votes
3answers
540 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 &...
4
votes
2answers
517 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 ...
8
votes
1answer
579 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'...
6
votes
1answer
263 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 ...
3
votes
1answer
209 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.
2
votes
1answer
239 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 ...
3
votes
0answers
63 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 ...
6
votes
1answer
465 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 ...
3
votes
1answer
209 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 $\mathbb{H}^{m\times m}$, ...
6
votes
1answer
164 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{...
3
votes
2answers
257 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 ...
9
votes
1answer
244 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_{\...
1
vote
0answers
74 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-...
5
votes
4answers
529 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 ...
1
vote
1answer
126 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 $$\...
2
votes
0answers
369 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 ...
6
votes
1answer
253 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 $\mathbb{A}$ ...
5
votes
2answers
226 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?
4
votes
1answer
253 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+...
4
votes
1answer
310 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 ...
3
votes
1answer
470 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 ...
1
vote
1answer
176 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 ...
1
vote
0answers
46 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 \\ ...
3
votes
1answer
192 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 $$ g_{st}{J^s}_i{J^t}_j=g_{ij}. \tag{1} $$...
2
votes
0answers
141 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....
2
votes
2answers
136 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\...
4
votes
1answer
701 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 ...
7
votes
2answers
1k 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 $q$...
5
votes
2answers
449 views

About some property of automorphism of octonions

Let $f$ be an automorphism of the algebra of octonions. Is it true that $f$ preserves some quaternionic subalgebra? Has the statement an elementary proof?
3
votes
1answer
209 views

Computing the Cartan1-form for $Sp(2)$

Context: I must find the Cartan 1-form for $Sp(2)$ before I start dealing with the natural connection of the Hopf fibration $S^3 \hookrightarrow S^7 \overset{\mathcal P}\to S^4$. To do so, the idea ...
21
votes
1answer
492 views

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 ...
4
votes
0answers
127 views

Exceptional symmetric spaces with quaternionic structure

Following this and this question I found following chain of exceptional symmetric spaces being quaternionic manifolds. I listed dimensions as superscripts for reader convenience. $F_{I}^{28}\subset ...
5
votes
1answer
283 views

Exceptional isomorphism with Spin(6,2)?

There are all sorts of curios in low-dimensional Lie groups and Lie algebras, many of them due to the presence of the quaternions. There is, I have recently learned, an isomorphism $SO(6,2) \simeq SO(...
4
votes
1answer
456 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 ...
1
vote
0answers
62 views

4-D lattices and quaternion

It is easy to prove that there are only 2 extensions $\mathbb{Q}(a)$, with $|a|=1$, of $\mathbb{Q}$ where $\mathbb{Z}[a]$ becomes a lattice(discrete free abelian subgroup of rank 2) in the complex ...
2
votes
0answers
208 views

A Quaternions version of the Gauss Lucas theorem

Edit: My serious thanks to Christian, Todd and Yemon for their comments to the previous version. The derivative is the same as Todd says: $(az^{n})'=naz^{n-1}$. The polynomial is in the form of $\...
11
votes
1answer
550 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. ...