# Questions tagged [quaternions]

The quaternions tag has no usage guidance.

123
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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} $ ...

5
votes

1
answer

136
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### 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 $\...

0
votes

1
answer

58
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### 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 + ...

3
votes

0
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63
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### 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 ...

8
votes

1
answer

217
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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 ...

1
vote

0
answers

102
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### 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\;,\...

1
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0
answers

87
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### 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 ...

3
votes

3
answers

369
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### 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 ...

2
votes

0
answers

62
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### 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 ...

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418
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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 ...

6
votes

4
answers

387
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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 ...

3
votes

1
answer

102
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### 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

1
answer

368
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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
$...

3
votes

1
answer

194
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### 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

1
answer

179
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### 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}^+...

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147
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### 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

0
answers

127
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### 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
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0
answers

85
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### 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

1
answer

236
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### 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

1
answer

205
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### 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

1
answer

248
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### 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

1
answer

234
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### 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

3
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563
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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 &...

4
votes

2
answers

551
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### 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

1
answer

641
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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'...

6
votes

1
answer

271
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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 ...

3
votes

1
answer

213
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### 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.

3
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2
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418
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### 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
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0
answers

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### 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

1
answer

516
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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 ...

3
votes

1
answer

244
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### 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

1
answer

180
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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{...

3
votes

2
answers

261
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### 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
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1
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280
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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_{\...

1
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0
answers

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### 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

4
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585
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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 ...

1
vote

1
answer

146
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### 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
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0
answers

443
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### 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

1
answer

284
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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 $\mathbb{A}$ ...

5
votes

2
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255
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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?

4
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1
answer

282
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### 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

1
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396
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### 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

1
answer

547
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### 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

1
answer

190
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### 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

0
answers

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### 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

1
answer

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### 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
$$
g_{st}{J^s}_i{J^t}_j=g_{ij}. \tag{1}
$$...

3
votes

0
answers

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### 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

2
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143
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### 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

1
answer

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### 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

2
answers

2k
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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 $q$...