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3
votes
0answers
140 views

Division algebras over extension fields / reducibility of $G$-modules

Reformulation of the question (see below for the original question): Let $K$ be an algebraic number field and $D$ a finite-dimensional $K$-division algebra. Is there a description of the field ...
10
votes
1answer
358 views

Can a division algebra have degree divisible by its characteristic?

I apologize in advance if this is easy, but I've tried Googling, and had no luck. I'm currently working on a proof, and I realized in the course of writing that this proof will break if out there ...
6
votes
0answers
173 views

mod $p$ Jacquet-Langlands correspondence

Let $F$ be a local field of characteristic $0$. Let $D$ be division algebra over $F$ of dimension $n^2$. The construction of irreducible complex representations of $D^*$ is known by Howe, Zink, and ...
0
votes
1answer
73 views

Regular or elliptic elements in the multiplicative group of central division algebra

For an element $g$ of a connected reductive group $G$ over a field $F$, $g$ is called $regular$ if the dimension of the centralizer of $g$ is equal to the rank of the algebraic group $G$, $g$ is ...
2
votes
0answers
65 views

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 ...
4
votes
1answer
163 views

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 ...
4
votes
0answers
262 views

Hermitian forms over quaternion algebra

Notations: Let $Q=(a,b)$ be a quaternion algebra over a field of characteristic $\neq 2$, i.e. $i^2=a, j^2=b, k=ij, ij=-ji$. Consider $K=k(t)(\alpha)$, where $\alpha=\sqrt{at^2+b}$. Let ...
4
votes
1answer
172 views

Is $SL_1(D)$ toplogically finitely generated, for $D$ a division algebra over a local field?

I've been struggling with this one all day, and I was wondering if someone can give me a hand with the proof. I'm not even sure if the group in question is finitely generated, so I would appreciate if ...
3
votes
2answers
595 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 ...
9
votes
1answer
445 views

Structure of units in a maximal order

Hello, my question is simple: do we have a "Dirichlet's unit theorem" for the group of units of a maximal order of a central division algebra ? In other words: let $k$ be a number field, let $D$ be ...
1
vote
2answers
320 views

What structure supports division to a unique quotient and remainder?

This has been bugging me for a while. According to https://en.wikipedia.org/wiki/Euclidean_division, if I divide integer $a$ by integer $b$, I get unique $t$, $r$ such that $a = t b + r$, $0 \le r ...
2
votes
1answer
167 views

equivalence of maximal fields in division algebras

Let D be a division algebra over F, E its maximal field. Is it true that: 1) all such fields are equivalent over F? 2) all such fields are conjugate by inner automorphisms of D?
12
votes
1answer
750 views

finite dimensional real division algebras

A celebrated theorem of Milnor and Kervaire asserts that any finite dimensional division algebra over the real numbers has dimension 1,2,4 or 8. This result is established using methods from algebraic ...
2
votes
1answer
204 views

(Non-)existence of skew fields satisfying a SGPI (=skew generalized polynomial identity)

Let $K$ be a skew-field, infinite dimensional over its center $F$. From Kaplansky's PI-theorem it then follows that $K$ cannot satisfy a polynomial identity (the theorem says that primitive ...
2
votes
1answer
573 views

Central division and quaternion algebras

I would like to know if there are some central-simple algebras $D_1$, $D_2$ and $D_3$ over a field $k$ satisfying the following properties : $ind(D_1)=exp(D_1)=4$ ($ind$ is the Schur index and $exp$ ...
12
votes
3answers
993 views

How to distinguish division algebras from matrix algebras?

Suppose $D$ is an explicitly given rank 9 central simple algebra over ${\mathbb Q}$ (or a number field). For example it could be specified by two cubic subfields ${\mathbb Q}(a), {\mathbb Q}(b)\subset ...
0
votes
1answer
607 views

identity for matrices whose determinant is 1.

For any matrices $A$,$B$ in $SL(2,\mathbb{C})$, let $M(A,B)$ be the 4 by 4 matrix whose columns are matrices $I$,$A$,$B$,$AB$. Then it is not hard to verify that $det M(A,B)= 2- tr[A,B]$. Is there an ...
16
votes
3answers
1k views

Infinite-dimensional normed division algebras

Let's say a normed division algebra is a real vector space $A$ equipped with a bilinear product, an element $1$ such that $1a = a = a1$, and a norm obeying $|ab| = |a| |b|$. There are only four ...
11
votes
2answers
687 views

units in distinct division algebras over number fields---are they definitely not isomorphic as abstract groups?

This is really an irrelevant question in the sense that the answer isn't remotely "logically crucial for the Langlands programme" or whatever---it's just something that occurred to me when writing ...
13
votes
4answers
971 views

Dimension of central simple algebra over a global field “built using class field theory”.

If $F$ is a global field then a standard exact sequence relating the Brauer groups of $F$ and its completions is the following: $$0\to Br(F)\to\oplus_v Br(F_v)\to\mathbf{Q}/\mathbf{Z}\to 0.$$ The ...
4
votes
3answers
712 views

Splitting of a division algebra with an involution of second kind

Let $k$ be a field, $K/k$ a separable quadratic extension, and $D/K$ a central division algebra of dimension $r^2$ over $K$ with an involution $\sigma$ of second kind (i.e. $\sigma$ acts non-trivially ...
-3
votes
4answers
2k views

Why don't quaternions have an overall phase? [closed]

The product of a quaternion multiplied by a real number is a quaternion, but the product of a quaternion multiplied by a complex number is not in general a quaternion. Why are the quaternions defined ...
2
votes
1answer
584 views

Maximal subfield inside a central division algebra

D is a central division algebra over F. We know that we can always find a maximal subfield K inside D such that K/F is separable. I want to know can we always make it Galois?