# Questions tagged [quaternions]

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

283 questions
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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!
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### 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 ...
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### 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)...
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### 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 (...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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,...
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### 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 $f:\mathbb{H} \rightarrow \mathbb{H}$?
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### 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 ...
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}{...