Questions tagged [quaternions]

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12 votes
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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 ...
BernyPiffaro's user avatar
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 ...
David Feldman's user avatar
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 ...
Dement's user avatar
  • 153
10 votes
0 answers
417 views

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 ...
user12806's user avatar
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6 votes
0 answers
362 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} $ ...
Max Muller's user avatar
  • 4,435
6 votes
0 answers
153 views

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 $...
Zurab Silagadze's user avatar
5 votes
0 answers
309 views

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 ...
Nicolas's user avatar
  • 51
4 votes
0 answers
219 views

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....
seub's user avatar
  • 1,337
4 votes
0 answers
141 views

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 ...
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3 votes
0 answers
91 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 ...
a196884's user avatar
  • 323
3 votes
0 answers
87 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 ...
Rybin Dmitry's user avatar
2 votes
0 answers
43 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}...
Giulio Binosi's user avatar
2 votes
0 answers
72 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 ...
Hvjurthuk's user avatar
  • 573
2 votes
0 answers
163 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 ...
Ali Taghavi's user avatar
2 votes
0 answers
164 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 ...
quaternion's user avatar
2 votes
0 answers
519 views

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 ...
user avatar
2 votes
0 answers
228 views

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 $\...
Ali Taghavi's user avatar
2 votes
0 answers
141 views

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 $F=\mathbb{Q}$)...
Darius Math's user avatar
  • 2,181
1 vote
0 answers
69 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 $GL(n, \mathbb{H})$ ...
Malkoun's user avatar
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1 vote
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30 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&...
Oleksandr  Kulkov's user avatar
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(\...
user484672's user avatar
1 vote
0 answers
120 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\;,\...
ql2000's user avatar
  • 11
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 ...
bobuhito's user avatar
  • 1,537
1 vote
0 answers
107 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, ...
BANOUH HICHAM's user avatar
1 vote
0 answers
51 views

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 \\ ...
Mtheorist's user avatar
  • 1,135
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 ...
user94475's user avatar
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 + \...
Mirco A. Mannucci's user avatar
0 votes
0 answers
211 views

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 ...
Shenghan Jiang's user avatar