# Tagged Questions

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### How to take partial derivative of spherical interpolation of quaternions?

Using the standard definition of quaternionic spherical linear interpolation (slerp): $$Q(q_0,q_1,t) := q_0(q_0^{-1}q_1)^t,$$ how can I take each partial derivative? Actually, I'm confident how to ...
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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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### 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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### Hurwitz integers represented as sums of two squares of Hurwitz integers

I wonder if there exists a characterisation of Hurwitz integers which are represented as sums of two squares of Hurwitz integers, up to multiplication by a unit. And if so, could you please point to a ...
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### What does the Jacquet-Langlands correspondence say about quaternion algebras of class number one?

If F is a totally real number field of degree n, and A is a definite quaternion algebra over F, I understand (not really) the Jacquet Langlands correspondence to construct a modular form in n ...
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### Which quaternion algebras have class number one?

Over Q, the definite quaternion algebras with a unique conjugacy class of maximal orders, i.e. "with class number one", are those with discriminant 2,3,5,7, and 13. Three questions: What is a ...
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### How do you find maximal orders in quaternion algebras?

Let A be the 4-dimensional algebra over Q with basis 1,i,j,k, and multiplication table $$i^2 = -1 \quad j^2 = -11 \quad k^2 = -11 \quad ij = k \quad jk = 11i \quad ki = j$$ So, A is the unique "...
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### Finding Finite Generators of a Subset of a Quaternion Algebra/Cocompact Lattices

I was wondering if anyone had some ideas (books, papers, experience) on how to explicitly compute generators for the elements of a quaternion algebra, $Q$, with reduced norm $1$. I'm trying to ...
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### How to calculate $(A^{-1})_{ii}$ for an invertible hyperhermitian quaternionic matrix $A$?

The article Alesker, S. (2003). Quaternionic Monge-Ampere equations. The Journal of Geometric Analysis, 13(2), 205-238. has the following CLAIM: Claim. Let $A$ be an invertible hyperhermitian ...
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### Question about the definition of the genus 0 curves in Gross' paper “Heights and the Special values of L-series”

Let $N \in \mathbb{Z}$ be a prime number, and let $B = \left( \dfrac{a, b}{\mathbb{Q}} \right)$ be the unique quaternion algebra over $\mathbb{Q}$ ramified at $N$ and at $\infty$. Then, in section 3 ...
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### Diagonalization of Quaternion Hermitian matrices

How do I go about diagonalizing such a matrix. I ask because I need to sort out the following problem: Let $D$ be the quaternion algebra over $\mathbb{Q}$ with $i^2 = -1, j^2 = -11, ij=-ji=k$. ...
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Have they been studied? In particular, what is the analogue of the Schmidt theorem for compact operators in Hilbert spaces? Helemskii A. Ya., Lectures and Exercises on Functional Analysis, Ch. 3, $\... 2answers 264 views ### Representing quaternions as matrices [closed] Assume F is a field of characteristic different than 2. Let a, b be invertible elements in F, and let A(a,b) be the generalised quaternions. Using the Artin–Wedderburn theorem, there is a ... 1answer 239 views ### Filling in a rational orthogonal matrix given one row Quick version: given natural$n$and a row of$n$integers such that the sum of the squares is another square, call it$m^2.$For$n=5,6,7$is it always possible to fill in the rest of an$n$by$n$... 2answers 246 views ### 3D Rotation Representation for Multiple Turns Is there a rotation representation that can also represent "turns", instead of collapsing coincident rotations into the same representation? In 2D, a simple angle satisfies this, as it can have ... 2answers 594 views ### ramified quaternion algebras I'm trying to better understand the connection between the concepts of ramification of a field extension, and ramification of a quaternion algebra. I'm also trying to build a better understanding of ... 2answers 735 views ### Skew fields inside quaternion division algebras Suppose that$Q$is a quaternion division algebra with center$k$, where$k$is an arbitrary commutative field (let's say with$\operatorname{char}(k) \neq 2$if necessary). Assume that$D$is an ... 3answers 1k views ### Realizing proper pure octonions as conjugates Let us take the octonions as having all integer coefficients and the multiplication table at BAEZ We have a standard conjugation operator with$\bar{1} = 1$and$\bar{e_i}= - e_i,$extend by ... 2answers 330 views ### Volumes of fundamental domains of maximal orders in definite quaternion algebras over Q I'm looking for an explanation of the following result: If D is a maximal order in a definite (i.e. ramified at infinity) quaternion algebra B over$\mathbb{Q}$, and$\phi : B\otimes\mathbb{R}\...
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 ...