Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [quaternions]

For questions about the quaternions: a noncommutative four dimensional division algebra over the real numbers. Also for questions about quaternion algebras.

11
votes
0answers
205 views

Does the exceptional Lie algebra $\mathfrak{g}_2$ arise from the isometry group of any projective space?

I learned from Baez's notes on octonions that the classical simple Lie algebras can be identified with the Lie algebras of isometry groups of projective spaces over $\mathbb{R}, \mathbb{C}$ and $\...
10
votes
0answers
572 views

Using Padé approximants for the quaternion exponential

On a lark, I wanted to know if one can use Padé approximants to compute the exponential $\exp(z)$ of a quaternion $z=a+b\mathbf{i}+c\mathbf{j}+d\mathbf{k}$. Since Mathematica has a package meant for ...
9
votes
0answers
367 views

Fractal derivative of complex order and beyond

Is there some precise definition of "complex (fractal) order derivative" for all complex number? I am aware of the Riemann-Liouville fractional definition given here: Complex derivative but I would ...
6
votes
0answers
86 views

Invariant homogeneous polynomial on quaternions

Let $\mathbb{H}$ denote the quaternions. If $(w_1,\ldots,w_n)\in \mathbb{H}^n$ we can write $w_i=z_i+jz_{n+i}$ with complex numbers $z_1,\ldots,z_{2n}$. Now let $M$ be the group of all matrices of the ...
5
votes
0answers
142 views

Trying to find a specific rotation quaternion

I'm looking for a way to find a specific rotation quaternion. Hoping that I'll get the notation right (no mathematician) the basic problem looks as follows Definitions $t \in \{0,...,n\}$ $R(Q, \...
5
votes
0answers
101 views

About definition of Fuchsian group derived from a quaternion algebra

In Katok's book, we have a defintion as following: If $\Gamma$ is a subgroup of finite index of some $\Gamma(A,\mathcal{O})$, then we call $\Gamma$ a Fuchsian group derived from a quaternion ...
5
votes
0answers
78 views

Hamilton's letter to his son

I'm looking for a better reference on this letter from Hamilton to his son where he wrote about his discovering on Quaternions. I'd like to read, if it is possible, a scanned version of the letter. ...
4
votes
0answers
93 views

Ideals of Modular Lipschitz Quaternions II: Progress and New Questions

I recently asked a queston regarding the proper, nontrivial ideals of what I called the modular Lipschitz quaternions, which was part of my series of open problems for enthusiasts. As luck would have ...
4
votes
0answers
86 views

Structure group of quaternionic manifold

Quaternionic manifolds, also called almost hypercomplex, are defined by the existence of an ($\mathbb{R}$-linear) action of quaternions on each tangent space such that $I,J,K\in\mathbb{H}$ are global ...
4
votes
0answers
72 views

4-D lattices and quaternions

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 ...
4
votes
0answers
235 views

Is there a way to prove vector triple product from quaternion multiplication?

For pure imaginary quaternions $u, v, w$, is there a way to prove the vector triple product $u\times(v\times w) = v(u\cdot w) - w(u\cdot v)$ from the relation: $$uv = -u\cdot v + u\times v \text{ for $...
4
votes
0answers
877 views

Are quaternions used in tensor analysis?

At the moment I am learning tensor analysis. In the books I study (civil engineering/mechanical books on tensors) there is a lot written about rotation matrices/tensors, however quaternions are not ...
4
votes
0answers
65 views

Constructing a coset representative of $SO(n,4)/(SO(n) \times SO(4))$.

In $\mathcal N = 2$ Supergravity the scalar components of Hypermultiplets form a quaternionic Kaehler manifold. Only isometries of this so-called target manifold can be gauged. I am interested in ...
4
votes
0answers
1k views

Tensor product of the spaces of quaternions and complex numbers

Let $ \mathbb{H} $ be the ring of quaternions and make the vector space $A = \mathbb{H} \otimes \mathbb{C}$ into a ring by defining $$(a \otimes w)(b \otimes z) = (ab \otimes wz) $$ for $a,b \in \...
4
votes
0answers
135 views

The geometry of $\operatorname{PSO}(4)$ and the quaternions

Question: Given a twist of the projective space, how do I find unit quaternions that represent it? Backgroud and what do I mean: Following Conway & Smith's On Quaternions and Octonions, every ...
3
votes
0answers
17 views

Elementary proof for even numbers of ramified places for a quaternion algebra over $\Bbb Q$

I want to explain to someone that for any quaternion algebra $H$ over $\Bbb Q$, the number of places of $\Bbb Q$ that $H$ is ramified is even. But he only knows basic of algebraic number theory, so I ...
3
votes
0answers
56 views

Complex and Quaternionic Matrices(Book)

I'm looking for a book on complex and quaternionic matrices and on the links between them and the real matrices. I'm reading the book Matrix grops for undergradtautes(K.Tapp) but I'm looking for ...
3
votes
0answers
57 views

Isomorphism between two generalized quaternions algebras

Suppose $p,q\in\mathbb{Z}$ are distinct primes that are congruent to $3\;(mod\; 4)$. I would like to prove that $$A=\left(\frac{-1,p}{\mathbb{Q}}\right)\not\simeq \left(\frac{-1,q}{\mathbb{Q}}\right)=...
3
votes
0answers
89 views

What is the logic behind proving there are infinitely many solutions (a quaternion problem)?

I am solving the following problem: $x\in\mathbb{H}$ where $x=a+bi+cj+dk$ and $a,b,c,d\in \mathbb{R}$ is called a pure quaternion if and only if $a=0$. Show that there are infinitely many pure ...
3
votes
0answers
83 views

Why can every two-dimensional left ideal in a quaternion algebra be written in this way?

Let us assume we are given a field $K$ and the quaternion algebra $Q=(a,b)_K$ with $a,b\in K^{\times}$. $Q$ is generated as a $K$-vector space by elements $1,i,j,k$ with $k=ij$, $i^2=a$, $j^2=b$ and $...
3
votes
0answers
57 views

Degree of quaternion product composed with two maps of $S^3$

Let $S^3$ denote the unit quaternions with multiplication $\mu:S^3\times S^3\rightarrow S^3$.Show that if $f_1,f_2:S^3\rightarrow S^3$ are given maps,that the composition $$S^3\xrightarrow{f_1\times ...
3
votes
0answers
443 views

Jacobian Matrix of 6DOF Body (with IMU)

I am trying to derive the analytical Jacobian for a system that is essentially the equations of motion of a body (6 degrees of freedom) with gyro and accelerometer measurements. This is part of an ...
3
votes
0answers
105 views

How stable is my quaternion interpolation?

after some experimentation to optimize slerp I found that finding the middle between quaternion is rather cheap (for $t=0.5$) in particular: (with $\theta$ the angle between $q_1$ and $q_1$) $$slerp(...
2
votes
0answers
31 views

Why convert Quaternion to Euler Angle

I've recently played with the IMU filter in MATLAB. When using their examples, they always plot the rotations by stating something alike this: ...
2
votes
0answers
24 views

How are these Quaternion Algebras isomorphic? $\Big(\dfrac{\mathit{a,b}}{\mathit{F}}\Big) \simeq \Big(\dfrac{\mathit{b,a}}{\mathit{F}}\Big)$

So let $\Big(\dfrac{\mathit{a,b}}{\mathit{F}}\Big)$ be a quaternion Algebra over a field $F$ with char $\neq 2$, and let $i,j$ be the standard generators for the quat. Algebra, meaning $i^2=a$ and $j^...
2
votes
0answers
53 views

Exterior Derivative over Quaternions

I was wondering if it is possible to define the exterior derivative of a quaternionic valued function. I am doing the quaternionic analogue of a previously complex valued computation, namely something ...
2
votes
0answers
43 views

$\mathbf{U}\times(\mathbf{V}\times\mathbf{W})=(\mathbf{U}\cdot\mathbf{W})\mathbf{V}-(\mathbf{U}\cdot\mathbf{V})\mathbf{W}$ Quaternion Proof

Prove the identity $$\mathbf{U}\times(\mathbf{V}\times\mathbf{W})=(\mathbf{U}\cdot\mathbf{W})\mathbf{V}-(\mathbf{U}\cdot\mathbf{V})\mathbf{W}$$ given three vectors $\mathbf{U},\mathbf{V}$ and $\mathbf{...
2
votes
0answers
91 views

How is quaternion analysis related to vector calculus?

I didn't really enjoy learning vector calculus in college, but I wanted to relearn the material. Like any reasonable person the first place I looked for resources was wikipedia, where I came across ...
2
votes
0answers
59 views

Generators for the special quaternionic orthogonal Lie algebra $\mathfrak{so}^*(2n)$

I'm searching for a reference of a "small" (close to minimal) set of generators for the (real) Lie algebra $\mathfrak{so}^*(2n)$, which can be defined as $$\mathfrak{so}^*(2n) = \left\{ \begin{pmatrix}...
2
votes
0answers
74 views

About commutator operator of the Quaternion algebra

Let $\mathbb K$ be a field of $\text{Char}$($\mathbb K$)$\neq 2$. Set $Q=Q(a,b\mid \mathbb K)=(a,b)_{\mathbb K}$ be the Quaternion algebra for $a,b \in \mathbb K$, with $\mathbb K$ basis $1, i, j$ and ...
2
votes
0answers
88 views

Why there are 48 distinct matrix representations available for quaternions?

According to Wikipedia, a Quaternion can be represent in 48 different $4\times4$ matrix forms. Finding this number can be proved by using permutation? $$\binom 4 2 \cdot4=\frac{4!}{(4-2)!}\cdot4=48$$ ...
2
votes
0answers
161 views

Logarithm and exponent of real quaternions

The logarithm of a general quaternion is defined as $$log(q) =\left (\left|q\right|, \frac{\mathbf{v}}{\left|\mathbf{v}\right|}cos^{-1}\left(\frac{r}{\left|q\right|}\right)\right),$$ in $(r,\mathbf{...
2
votes
0answers
62 views

Is every finite group an arithmetic group?

Lubotzky-Phillips-Sarnak(LPS88) constructed Ramanujan graphs of degree $d=p+1$ (for an odd prime $p$) as Cayley graphs of projective linear groups with respect to a carefully chosen set of generators ...
2
votes
0answers
34 views

Is there any articles about rigid body full state integration and linearization in terms of quaternions?

I'm trying to develop a control for a quadrotor, but I'm struggling with position and velocity stabilization. I believe that I have an issue with position and velocity states of the system. Maybe ...
2
votes
0answers
98 views

Is the product of two Guassian quaternions a Gaussian quaternion?

Suppose we have $q_1\sim \mathcal N(\mu_1,\Sigma_1)$ and $q_2\sim\mathcal N(\mu_2,\Sigma 2)$ two independent Gaussian Random Variables that represent two quaternions. Consider: $$ q_3 = q_1\otimes q_2 ...
2
votes
0answers
91 views

Complexification of a field

I admit that the term "complexification" is rather ill chosen and I'll gladly see it replaced by any other denomination. The term occured to me in the context where I had the impression that the ...
2
votes
0answers
796 views

Quaternion - switching from right-handed to left-handed

I have a rotation quaternion in my 3D model and it uses the right-handed coordinate system. Now I'd like to convert it into a left-handed since my game engine (Unity) uses left-handed. These are the ...
2
votes
0answers
78 views

Quaternion structure implies dimension divisible by $4$

A quaternion structure on a real vector space $V$ is a pair of operators $J,K \in \mathcal{L}(V)$ such that $$J^2 = K^2 = -I, \; \; JK = -KJ$$ where $-I$ is the negative identity matrix. This makes $...
2
votes
0answers
50 views

Hamilton's Lectures on Quaternions or Elements of Quaternions

Does anyone know of any detailed study (book, dissertation, thesis, etc.) on Hamilton's Lectures on Quaternions? I've tried several places but I have not found anything yet. Thanks any help!
2
votes
0answers
357 views

Why are quaternions (over the field of complex numbers) two-rowed matrix algebras?

I've enjoyed D.E. Littlewood's Skeleton Key to Mathematics up to chapter 8. After chapter 8, increasingly more statements are left without proof and I'm lost. He writes [...]it can be shown that ...
2
votes
0answers
245 views

Is there a residue theorem for Quaternions?

One of Complex Analysis's biggest contributions is the residue theorem. Is there a similar theorem in the field of Quaternion Analysis? (A glance at Wikipedia didn't pull anything that caught my eye)...
2
votes
0answers
102 views

Factoring quaternion into three parts

I am faced with a problem where I have a quaternion which represents rotation and three arbitrary axis about which I can make rotations, thus three unit vectors. What I would like to know is angles (...
2
votes
0answers
58 views

Grothendieck group of a quaternion algebra

Let $\mathcal{O}$ be a maximal order in a quaternion algebra over a number field. Then there is a notion of similarity of (left) $\mathcal{O}$-modules, and similarity classes. The set $S$ of such ...
2
votes
0answers
113 views

Explaining Spin(3)

I’m going to discuss the action of Spin(3) on Euclidean vectors. This thing has several alternative names: “versors”/“rotation quaternions”, “quaternionic adjoint representation”, “quaternion action ...
2
votes
0answers
544 views

Quaternion Calculus

I was reading a note on Quaternion(Link) and I am happened to read a section regarding a solution of quaternion differential equation. I am putting that segment as picture format here for more ...
2
votes
0answers
102 views

Strong Approximation for adelic quaternionic groups

Let $H$ be a definite quaternion algebra defined over $\mathbb{Q}$, unramified at $p$, and ramified at $\infty$. Denote by $D$ the multiplicative group $H^\times$ divided by its center. In this paper,...
2
votes
0answers
75 views

Inner product space or Hilbert space of Quaternionic Functions

In what ways can you define an inner product, $<f,g>$, to create an inner product space or Hilbert space on the set of quaternionic functions $f:\mathbb{H} \rightarrow \mathbb{H}$?
2
votes
0answers
108 views

Does Shor's algorithm work for noncommutitive or nonassociative algebras?

Shor's algorithm is said to solve the hidden subgroup problem for finite Abelian groups, at least according to the Wikipedia article I read. This means that it can be used to solve factoring or ...
2
votes
0answers
363 views

Dual quaternion derivation

I'd like to derivate a dual quaternion \begin{align} \hat{q}&=(1 + \frac{1}{2}\epsilon\vec{t})q \end{align} where \begin{align} q &= e^\vec{w} , \\\vec{w}&=(0, w_1,w_2,w_3)^t \end{...
2
votes
0answers
40 views

Computing a particular finite set of quaternion matrices.

Let $B = \left(\frac{-1,-11}{\mathbb{Q}}\right)$ be a choice of quaternion algebra ramifying at $11$ and consider the maximal order $\mathfrak{O}=\mathbb{Z}\oplus\mathbb{Z}i\oplus\mathbb{Z}\frac{1+j}{...