Questions tagged [quaternions]
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142 questions
0 votes
0 answers
91 views
Line element for dual quaternions
I've been playing with dual quaternions as they relate to rigid transformations and I have some questions. First the background information. A dual number in this case is referring to a number system ...
0 votes
0 answers
97 views
The group of $p$-unit quaternions
Let $p$ be a prime and consider the ring $R=\mathbb{H}(\mathbb{Z}[\frac{1}{p}])$ of quaternions over $\mathbb{Z}[\frac{1}{p}]$. I am interested in the group $G=R^{*}/\{p^n,n\in\mathbb{Z}\}$. The ...
- 151
1 vote
1 answer
178 views
Hilbert symbol of a quaternion algebra given ramified places
I am reading the paper: https://projecteuclid.org/journals/experimental-mathematics/volume-17/issue-3/Derived-Arithmetic-Fuchsian-Groups-of-Genus-Two/em/1227121388.full in order to find an explicit ...
- 155
8 votes
0 answers
230 views
Bhargava's "Higher composition laws V" - where I can find it?
TSIA. There are four papers by Bhargava on higher composition laws that are publicly available: I: A new view on Gauss composition, and quadratic generalizations II: On cubic analogues of Gauss ...
- 2,245
2 votes
0 answers
156 views
gcrd and associates of an element of the quaternion algebra over a totally real number field $K$
Let $K$ be a totally real number field of class number 1, and $Q$ the quaternion algebra over the ring of integers of $K$ with basis $\{1,i,j,k\}$ such that $i^2 = j^2 = k^2 = -1$ and $ij = -ji, ik = ...
- 133
1 vote
1 answer
209 views
Quotient rings of integral quaternion rings
I'm having a hard time finding information about the quotient rings of the Lipschitz quaternions and the Hurwitz quaternions. The Lipschitz quaternions are defined as the quaternions with integral ...
- 1,352
1 vote
0 answers
126 views
Is there a quaternionic analogue of Weyl's character formula?
I am pondering about this. The definition of a character makes sense for quaternionic matrices. Indeed, given a quaternionic representation of a quaternionic matrix group such as $GL(n, \mathbb{H})$ ...
- 5,357
4 votes
2 answers
428 views
Lifting of map from $S^3$ to itself
My question concerns the lifting of degree $0$ map from $S^3$ to itself. Let us suppose that all maps are smooth here. Looking at $S^3$ as the space of unit quaternions, one way to define degree is ...
- 742
2 votes
0 answers
76 views
Geometric explanation of Fueter-Sce-Qian Theorem and similar situations
In Clifford analysis there is a fundamental theorem due to Fueter and extended by Sce and Qian that says (in modern terminology) that the given a slice regular function $f:\mathbb{R}^{m+1}\to\mathbb{R}...
- 131
1 vote
0 answers
55 views
Eigendecomposition of hyper-complex multiplication
There is an isomorphism between quaternions and $4\times 4$ matrices: $$ \phi: a+bi+cj+dk \longmapsto \begin{pmatrix} a&b&c&d \\ -b&a&-d&c\\ -c&d&a&-b\\ -d&-c&...
- 1,211
3 votes
2 answers
1k views
Is there a definition of $\log(x)$ for quaternion/octonion $x$?
I'm trying to implement $\log({\bf q})$ in python, where ${\bf q} = (q_0,\ldots,q_7) \in \mathbb{O}$ is an octonion. There is a well known definition of $\log({\bf q})$ for quaternions ${\bf q} = (s,v)...
- 2,241
4 votes
1 answer
258 views
Subfields of division rings of degree $2$ which are not invariant
Let $A$ be a noncommutative division ring, and let $B$ be a sub division ring (here, $B$ is allowed to be commutative) of degree $2$. Are there easy examples known for which $B$ is not globally fixed ...
- 4,819
12 votes
0 answers
473 views
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 ...
- 149
2 votes
1 answer
160 views
Polar decomposition with respect to the nonstandard involution of quaternionic matrices?
The quaternions admit infinitely many involutions. But up to isomorphism, there are only two: The standard one $t+xi+yj+zk\mapsto t-xi-yj-zk$ and the nonstandard one $\phi:t+xi+yj+zk\mapsto t-xi+yj+zk$...
- 5,023
27 votes
5 answers
4k views
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 "...
- 287
4 votes
1 answer
352 views
Abelianization of unit quaternions over a p-adic field
Suppose $p$ is a prime, that $F$ is a finite extension of the field $\mathbb{Q}_p$, $D$ is the division quaternion algebra over $F$ and $\mathcal{O}_D$ is the valuation ring of $D$. What is the ...
- 3,604
5 votes
1 answer
532 views
Motivating unpublished statements of Gauss about congruences and quaternions
Background Modern claims that Gauss anticipated the quaternions algebra are based primarily on an unpublished fragment of Gauss dated to 1819 and entitled "rotations of space". In this ...
- 2,497
1 vote
0 answers
468 views
Heat kernel on quaternion Heisenberg group
For the n Heisenberg($\Bbb C^n\times\Bbb R$) it is known that the heat kernel $q_s(z,t)=c_n\int_{\Bbb R} e^{-i\lambda t}\Big( \frac{\lambda}{\sinh(\lambda s)}\Big)^n e^{-\frac{\lambda|z|^2\coth(\...
- 11
2 votes
4 answers
941 views
How to prove that a quaternion algebra over ℤₚ is isomorphic to Mat₂(ℤₚ) for p prime?
How to prove without using advanced theorems that quaternions algebra $H = \genfrac(){}{}{-1,-1}{\mathbb{Z}_p}$, where $p$ is prime that $H \cong\operatorname{Mat}_2({\mathbb{Z}_p})$? My ideas: I ...
6 votes
0 answers
453 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} $ ...
- 5,055
5 votes
1 answer
225 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 $\...
- 39.4k
2 votes
1 answer
142 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 + ...
- 1,567
3 votes
0 answers
117 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 ...
- 323
8 votes
1 answer
318 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 ...
- 847
1 vote
0 answers
140 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\;,\...
- 11
1 vote
0 answers
111 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 ...
- 1,567
3 votes
3 answers
1k 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 ...
- 921
2 votes
0 answers
99 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 ...
- 583
16 votes
1 answer
1k 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 ...
- 367
6 votes
4 answers
791 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 ...
- 619
3 votes
1 answer
191 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 ...
- 338
10 votes
1 answer
630 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 $...
- 24.5k
3 votes
1 answer
517 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^{...
- 117
3 votes
1 answer
574 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}^+...
- 163
2 votes
0 answers
183 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 ...
- 290
2 votes
0 answers
225 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 ...
- 61
1 vote
0 answers
150 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, ...
7 votes
1 answer
674 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 ...
- 2,497
2 votes
1 answer
332 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 \...
- 2,602
1 vote
1 answer
316 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 ...
- 19
0 votes
1 answer
556 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 $x$ ...
- 1,901
8 votes
3 answers
723 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 &...
- 3,735
4 votes
2 answers
656 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 ...
- 5,023
9 votes
1 answer
970 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'...
- 71.2k
6 votes
1 answer
323 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 ...
- 1,593
3 votes
1 answer
254 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.
- 1,901
3 votes
2 answers
531 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
0 answers
114 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 ...
- 645
6 votes
1 answer
666 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 ...
- 517
3 votes
1 answer
485 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 $\mathbb{H}^{m\times m}$, ...
- 3,269