Questions tagged [quaternions]
The quaternions tag has no usage guidance.
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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 $$\...
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Symplectic group over finite field and quaternions
Symplectic group over finite field is defined as group preserving non-degenerate antisymmetric bilinear form on $\mathbb F_q^{2n}$. How could we define this group using quaternions ? This should be ...
user21230
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Computing Tamagawa number of torus in Quaternion algebra
Consider a rational Quaternion algebra $M$ over $\mathbb{Q}$ that does not split at $\infty$. For example take the rational Hamilton quaternions $M=\mathbb{Q}(-1,-1)$.
For the adele ring $\mathbb{A}$ ...
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Automorphic quotient for quaternion algebras
Are automorphic quotient for quaternion algebras always compact (safe the totally split case)?
Is there any good reference for proof of this fact, or easy arguments to say do?
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Why is this mapping surjective?
It is mentioned in wikipedia that every single orthogonal $3 \times 3$ rational matrix is of the form
$$\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+...
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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 ...
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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 ...
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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 ...
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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
$$
J_1= \left[\begin{array}{cccc}
0 & -1 & 0 & 0 \\
1 & 0 & 0 & 0 \\
...
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How does the concept of a hermitian metric generalize to a hyperkahler manifold?
A complex manifold admits an almost complex structure, $J$, which satisfies
$$
{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}
$$...
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Terminology: Almost hyper-Hermitian vs Almost hyper-Kähler vs
After non-thorough literature search, it seems to me that there is no consensus on the usage of the terminology "(almost) hyper-Hermitian" vs "almost hyper-Kähler" vs "almost quaternion-Hermitian" etc....
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Matrix expression for elements of $\text{SO}_0(1,4)$
Denote by $\text{SO}_0(1,4)$ the identity component of the special linear isometry group $\mathrm{SO}(1,4)$ of the Lorentz-Minkowski space $\mathbb{R}_1^5$, that is, of
$$\text{SO}(1,4)=\left\{X\in\...
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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 ...
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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 $q$...
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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?
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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 ...
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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 ...
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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 ...
user21230
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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(...
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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 ...
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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 ...
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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 $\...
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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.
...
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Circle Action on Quaternionic Projective Space
Quoting from Wikipedia article on quaternionic projective space:
Therefore the quotient manifold
$$
\mathbb{HP}^{2}/\mathrm{U}(1)
$$
may be taken, writing $U(1)$ for the circle group. It has ...
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Does Feuter regularity imply derivability in all directions?
The standard type of regularity in Clifford Calculus is the one introduced by Feuter, namely:
a function is Feuter regular iff it is in the zero set of the Clifford-Dirac
operator $D= \partial x_0 + \...
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Quaternion orders such that every proper ideal is invertible
Let $B$ be a quaternion algebra over $\mathbb{Q}$ and let $\mathcal{O} \subset B$ be an order.
A lattice in $B$ is (left) proper over $\mathcal{O}$ if its left order is equal to $\mathcal{O}$. We ...
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4th Order Floretions: Floret's Equation [closed]
Update: I've marked this question as answered. If you are thinking "What the heck are floretions?", go right to the answer provided by the Grinch. I definitely should have added clearer information on ...
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Hochschild cohomology of SU(2)
I have a question about the computation of an Hochschild Cohomology. Or at least about a space which really looks like a cohomology space. All the functions i consider are assumed to be smooth.
Let's ...
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Quaternions: ellipse effect
I would be interested in an explanation of the "six ellipse effect" produced by the pseudocode below (I also wonder how close are these to being actual ellipses). Note the code is somewhat similar to ...
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Is an associative division algebra required for this phenomenon?
For which integers $d \geq 1$ can we find real matrices $R_1, \dotsc, R_d$ of size $d \times d$ such that for any unit vector $v \in \mathbb{R}^d$, $$R_1 v, \dotsc, R_d v$$ is an orthonormal basis? ...
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Factoring quaternion into three parts [closed]
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 (...
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Classification of symplectic representations of quaternion division algebras
I would like to know the classification of representations of the form $\rho:B^{\times}\to Sp(V,F)$ or ($Gsp(V)$), where $B$ is a quaternion division algebra over a number field $F$ (or $F=\mathbb{Q}$)...
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Do unit quaternions at vertices of a regular 4-simplex, one being 1, generate a free group?
Choose unit quaternions $q_0, q_1, q_2, q_3, q_4$ that form the vertices of a regular 4-simplex in the quaternions. Assume $q_0 = 1$. Let the other four generate a group via quaternion ...
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On conductors, levels and traces on quaternion algebras
I am currently working on level issues in the division central simple algebra case, say $D$ over a local non-archimedean field $F$ (e.g. $\mathbf{Q}_p$). Let say that $\mathcal{O}_D$ and $\mathcal{O}...
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Maximal compact subgroup of $\mathrm{SL}(2,\mathbb{H})$
$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\SL{SL}$The exceptional isomorphism $\Spin(5,1)\simeq \SL(2,\mathbb{H})$ is well-known, and I can find references that ...
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Is the "algebraic closure" of the quaternions, finite dimensional? [closed]
This post is a sequel of: What's the algebraic closure of the quaternions?
$\mathbb{H}$ is algebraically closed for the polynomials of the form $\sum a_r x^r$, but it is not for the polynomials ...
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Canonical forms of symmetric/skewsymmetric quaternionic matrix
$A$ belongs to $n$-dimensional quaternion symmetric matrix, in the sense that $A=A^T$, where $T$ means transpose. Under transformation $U$, $A\rightarrow U\cdot A\cdot U^T$, where $U$ is $n$-dim ...
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Number of orbits of $\mathrm{SL}_2(\mathcal O_A)$ on $\mathbf P^1(A)$ when $A$ is a quaternion algebra
This is a reference request.
Let $A$ be an anisotropic quaternion algebra over $\mathbf Q$. Let $\mathcal O_A$ be a maximal order in $A$. Then $\mathrm{SL}_2(A)$ acts transitively on the right on ...
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How can the Cayley table for the elements of basis of a Cayley-Dickson algebra be summarized in an algebraic expression?
One would be able to construct a Cayley table that has all $e_i$ elements of the basis of algebra $A$ where $0<i<\dim A$ such that $e_0=1$, $e_1=i$, $e_2=j$ and so on. I'm looking for an ...
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Factorisation of local quaternionic zeta functions
Vignéras, in her Arithmetics of quaternion algebras, begin chapter II.4 recalling that we know the number of integer ideals of fixed norm of a quaternion algebra $H$ over a local field $K$, ramified ...
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Expression and growth bound for $r_{p^m,k}(n)$
Let's define , $$R_{p^m,k}(n)=\#\{(a_1,\dots,a_k)\in\mathbb{Z}^k:\sum_{i=1}^ka_i^2\le n \ \text{and} \ p^m|\sum_{i=1}^ka_i^2\}$$
what will be growth bound of $R_{p^m,k}(n)$? This can be thought as a ...
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Why do noncocompact arithmetic Kleinian groups have quadratic trace fields?
I realize there are a few different ways of going about proving this, depending on one's background, but there's a particular number theoretic aspect that I am just blanking on, and can't seem to find ...
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Which finite nonabelian groups have all their quaternionic representations of degree one?
A finite group $G$ has a finite set of irreducible representations over the complex numbers. All of these representations are linear (that is, are maps in 1x1 complex matrices) if and only if $G$ is ...
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Find a common expression (parameterized by y = 1/2, 1 and 2) uniting three (rational) polynomials in k [closed]
I am seeking a common "simple" expression (preferably/presumably a sum of products of linear factors, or sum of products of low-degree factors) uniting the three polynomials (parameterized by $y=\...
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Grunsky-Motzkin-Schoenberg formula
I found this formula in Brian McCartin's interesting book "Mysteries of the equilateral triangle" http://www.kettering.edu/news/mysteries-equilateral-triangle and it looks as follows:
Suppose that $...
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Explicit isomorphism for quaternion algebras over $\mathbb{Q}$?
It is known that the isomorphism class of a quaternion algebra $A=\binom{a,b}{K}$ over a number field $K$ is determined by the finite set of places $v$ of $K$ where $A\otimes_K K_v$ is a division ...
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Integral elements of quaternion algebras with predescribed properties
In the course of doing some calculations I have found myself wanting to answer the following question:
Let $D/\mathbb{Q}$ be a quaternion algebra ramified at a prime $p$ and at $\infty$ and let $\...
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history of quaternion algebras
Who is responsible for the generalization of Hamilton's quaternions to other types of quaternion algebras, and when did this occur? In particular, Hamilton's quaternions are the 4-dimensional algebra ...
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The de Rham complex of a quaternion-Kahler manifold
As we all know, for a complex manifold $M$, its de Rham complex admits a decomposition into a double complex called the Dolbeault complex. If $M$ also admits a Kahler metric, then we get the wonderful ...
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riemann mapping theorem for skew-fields of quaternions and beyond
Is there any known generalization of the riemann mapping theorem over skew-fields of quaternions and beyond or at least a conjectured formulation of it?
In a less focused way, how far does the main ...
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