Questions tagged [quaternions]
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28
questions with no upvoted or accepted answers
12
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418
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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 ...
12
votes
0
answers
441
views
Nullstellensatz for quaternionic plane curves?
By a quaternionic plane curve I mean the zero locus of a noncommutative polynomial in two variables, $x$ and $y$ say, over ${\Bbb H}$, Hamilton's quaternions. It is evidently well-known that, after ...
10
votes
0
answers
634
views
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 ...
10
votes
0
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417
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Polynomial function from $S^3$ to $S^3$ and quaternions
I am searching the polynomial functions from $S^3$ to $S^3$.
($S^3$ is the set of vectors $x$ in $\mathbb{R}^4$ such that $\|x\|=1$)
We say $g$ is a polynomial function from $S^3$ to $S^3$, if there ...
6
votes
0
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362
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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} $ ...
6
votes
0
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153
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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 $...
5
votes
0
answers
309
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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 ...
4
votes
0
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219
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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....
4
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141
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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 ...
3
votes
0
answers
91
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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 ...
3
votes
0
answers
87
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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 ...
2
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43
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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}...
2
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72
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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 ...
2
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answers
163
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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
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0
answers
164
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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 ...
2
votes
0
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519
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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 ...
2
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228
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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 $\...
2
votes
0
answers
141
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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 $F=\mathbb{Q}$)...
1
vote
0
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69
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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 $GL(n, \mathbb{H})$ ...
1
vote
0
answers
30
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Eigendecomposition of hyper-complex multiplication
There is an isomorphism between quaternions and $4\times 4$ matrices:
$$
\phi: a+bi+cj+dk \longmapsto \begin{pmatrix}
a&b&c&d \\
-b&a&-d&c\\
-c&d&a&-b\\
-d&-c&...
1
vote
0
answers
438
views
Heat kernel on quaternion Heisenberg group
For the n Heisenberg($\Bbb C^n\times\Bbb R$) it is known that the heat kernel $q_s(z,t)=c_n\int_{\Bbb R} e^{-i\lambda t}\Big( \frac{\lambda}{\sinh(\lambda s)}\Big)^n e^{-\frac{\lambda|z|^2\coth(\...
1
vote
0
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120
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Conway's quaternion notation +1/3[C_3×C_3]∙2^(2) represent C_3h of 4d Point group?
In John Conway and Derek Smith's On Quaternions and Octonions: their Geometry, Arithmetic, and Symmetry, they introduce a way to connect quaternions to 4D Point Group.
Suppose: $[l,r]:x\to \bar lxr\;,\...
1
vote
0
answers
88
views
Sequences generated from commuted quaternions and general commuted linear transformations
Given a pair of non-commuting linear transformations, $A$ and $B$, define the "next
pair" in a sequence as $A*B$ and $B*A$. I am interested in finite cycles (i.e.,
the sequence eventually ...
1
vote
0
answers
107
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Control and observability of Clifford algebra and quaternion valued systems?
Good evening everybody, I'm asking you if there is a mathematical theory about control and observability of differential systems within the quaternionic and Clifford (geometric) algebras. And if so, ...
1
vote
0
answers
51
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Which hyperkaehler manifolds admit an atlas with affine overlap maps?
In order for the quaternionic structure on a hyperkahler manifold to take the canonical form
$$
J_1= \left[\begin{array}{cccc}
0 & -1 & 0 & 0 \\
1 & 0 & 0 & 0 \\
...
1
vote
0
answers
72
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 ...
1
vote
0
answers
142
views
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 + \...
0
votes
0
answers
211
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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 ...