# Questions tagged [division-algebras]

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62
questions

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votes

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103 views

### The size of endomorphism rings and the relation to ordinariness of Abelian surfaces

For Elliptic curves over a finite field, there is a very useful characterization of ordinary elliptic as those with commutative, quadratic endomorphism rings and of supersingular curves as those with ...

**3**

votes

**0**answers

59 views

### On noncommutative transcendence degrees

The original transcendence degree for (noncommutative) division algebras is the Gelfand-Kirillov transcendence degree, due to I. Gelfand and K. Kirillov ([ Sur les corps li´es aux algèbres ...

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72 views

### On the Gelfand-Kirillov Conjecture

The base field $k$ is of zero characteristic.
Notation: $A_{n,s}(k):= A_n(k(x_1,\ldots,x_s))$, the Weyl agebra over a purely transcedental extension of the base field; $F_{n,s}(k)$, the Weyl field, is ...

**1**

vote

**1**answer

115 views

### Lattices from quaternion algebras (MAGMA software)

I am studying the paper "Lattice Packing from Quaternion Algebras" from 2012 about the construction of ideal lattices.
In Section 3.3 the authors construct very interesting examples of lattices using ...

**3**

votes

**0**answers

47 views

### A rationality question over skew-fields

Let $K$ be a skew-field, $k$ its center, and $V$ a finite-dimensional left vector space over $K$. Assume that $E$ is a $k$-subalgebra of $\mathrm{End}_K(V)$, and that $E$ is a (commutative) field.
...

**2**

votes

**1**answer

108 views

### Field of definition of a finite dimensional division algebra and how to reduce it

Let F be a field, and E/F an infinite algebraic extension. Let D be a finite dimensional division algebra over E (meaning its center is also E).
Is it possible to somehow gow down to a finite ...

**2**

votes

**1**answer

173 views

### Schur index of a representation and its divisors

We fix following objects:
(1) $G$ is a finite group.
(2) $\chi$ is complex irreducible character of $G$.
(3) $m$ is the Schur index of $\chi$ w.r.t. the rational field $\mathbb{Q}$.
(4) All the ...

**3**

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**0**answers

100 views

### Free skew fields over sets of different cardinal

Let $K$ be a field and let $X$ be a set. Denote by $\mathcal D_K(X)$ the free skew $K$-field on $X$.
Assume that $|X|\ne |Y|$. Is it true that $\mathcal D_K(X)$ and
$\mathcal D_K(Y)$ are not ...

**3**

votes

**1**answer

134 views

### Infinite dimensional finitely generated algebraic division algebra

Is there a division algebra $D$ with center $K$ that satisfies the
following 3 conditions?
1) $D$ is of infinite dimension over $K$;
2) every element of $D$ is algebraic over $K$;
3) $D$ is ...

**4**

votes

**1**answer

61 views

### Left vs right degree of skew-field extensions

Artin in his book, Geometric Algebra, says the connection between the left
degree and right degree of a skew-field extension is unknown.
Since I'm not an expert, I was wondering if someone knew the ...

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votes

**0**answers

129 views

### Projective modules over maximal orders of central simple algebras

In "Supersingular K3 surfaces" by TetsuJi Shioda, when proving Theorem 3.5 (Deligne) he considers a supersingular elliptic curve $C$ over an algebraic closed field of $\text{char}\ p>0$ and let $R ...

**0**

votes

**1**answer

179 views

### Classification of finite-dimensional real super C*-algebras

The title says it all. I feel like one should be able to find this somewhere, but every time I try to google, I just get results for "super Lie algebras". Does anybody know a reference? I am not so ...

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votes

**1**answer

188 views

### Image of the norm map for degree $3$ galois extension over $\mathbb{Q}$

I want to construct a cyclic division algebra of degree $3$ over some degree $3$ Galois extension $E$ of $\mathbb{Q}$. So the construction is as follows:
As a set $D=E\oplus uE \oplus u^2 E$ where $u$...

**6**

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**0**answers

195 views

### Unital nonalternative real division algebras of dimension 8

Cross-posted from Math SE because I felt like it might be too obscure for there. I'm sorry if this is the wrong place for it.
EDIT: This question now has an answer over there
The finite-dimension ...

**4**

votes

**0**answers

105 views

### 3-torsion part of Brauer group

I want to solve this problem:
If in field $K$ we have sufficient n-th roots of unity then the 3-torsion part of Brauer group is generated by classes of cyclic algebras
I know that every element in 3-...

**2**

votes

**1**answer

134 views

### Hermitian forms over $K\times K$

Let $V$ be a finitely generated module over the ring $R=K\times K$ where $K$ is a field. We fix the switch involution on the ring $R$. Let $H$ be a hermitian form over $V$.
When $V$ is a free module, ...

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61 views

### Spliting of division algebras over a ring

Let $D$ be a central division algebra over $\mathbb{Q}_p$, of dimension $n^2$. For example, let $D=E_n(\Pi)$, where $E_n$ is the unramified degree extension of $\mathbb{Q}_p$ such that $\Pi^n=p$ and $\...

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247 views

### Finding a cyclic cubic extension of a field

Let $K$ be a field and let $E/K$ be a Galois extension of degree 6 with $\text{Gal}(E/K) = S_3$, the symmetric group on 3 letters. Pick two different transpositions $s_1, s_2$ in $S_3$ (hence $s_1s_2$ ...

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46 views

### Isotropy of skew-Hermitian forms over division algebras

Assume char(F) $\neq$ 2.
Let $D$ be a central division algebra over a field $F$ and $h: V \rightarrow D$ be an anisotropic skew-Hermitian form. We can easily see that $h_{\bar{F}}$ is totally ...

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votes

**2**answers

156 views

### When is the following fraction an integer $\frac{3^a}{2^b-3}[(\frac{2^b}{3})^c - 1]$ where $a,b,c \in \mathbb{Z}$? [closed]

I'm trying to evaluate the following fraction $\frac{3^a}{2^b-3}[(\frac{2^b}{3})^c - 1]$, but I'm getting stuck using gcd arguments or divisibility arguments. This is part of an ongoing research I'm ...

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80 views

### Is $x \in A_1$ left algebraic over the subalgebra generated by $p$ and $q$, $[q,p]=1$?

Let $A_1:=A_1(x,y,k)$ be the first Weyl algebra over a field $k$ of characteristic zero,
namely, the $k$-algebra generated by $x$ and $y$ with relation $yx-xy=1$.
Let $f:(x,y) \mapsto (p,q)$ be a $k$-...

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120 views

### Real endomorphism algebra of abelian surface is never $\mathbb{C}$?

I'm reading about the Sato Tate conjecture for genus 2, and I came to the paper here. This breaks the conjecture into 6 different parts, based on the real endomorphism algebra of the surface, or the ...

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votes

**1**answer

218 views

### Octonion algebras over $\mathbb{F}_p(t)$

In their book Octonions, Jordan Algebras and Exceptional groups
Springer and Veldkamp have a subsection called 'Classification over special fields' in which they describe the number of division and ...

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votes

**1**answer

105 views

### Idea of base change for Division algebras over local field

Let $F$ be a non-Archimedean local field of characteristic $0$ and $K/F$ be a finite extension. Let $D_F$ be the central division algebra of dimension $n^2$ over $F.$ Write $D_K=D_F\otimes_FK$, which ...

**5**

votes

**1**answer

174 views

### Endomorphism algebras of restricted representations

Let $G$ be a group, and
$$\rho:G\to \mathrm{GL}(V)$$
be an absolutely irreducible, finite-dimensional representation over a characteristic $0$ field $k$. For each finite index subgroup $H\le G$, let
$...

**8**

votes

**2**answers

215 views

### Representations of $SL_1(D),$ where $D$ a division algebra over a local field

Let $k$ be a local field of residue characteristic $p$, and let D be a central
division algebra over $k$ of index $n>2$. How to determine the irreducible complex representations of the group $SL_1(...

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221 views

### Dimension of the moduli space of abelian varieties with a prescribed endomorphism algebra

Let $D$ be a division algebra over a number field $K$, and consider abelian varieties $A$ over the complex numbers, of dimension $g$ with an action of (an order of) $D$. Is it known when this set is ...

**7**

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**1**answer

478 views

### When is $GL_m(R)$ generated by elementary and diagonal matrices?

Let $D$ be a division ring and $R=D[t_1,\ldots,t_n]$ the polynomial ring in $n$ variables. Now let $GL_m(R),\,E_m(R)$ be the usual general linear group and its subgroup generated by the elementary ...

**2**

votes

**1**answer

290 views

### Dimension of maximal tori in division algebras

Does there exist a field $K$ and a finite-dimensional $K$-division algebra $D$ possessing two maximal separable subfields of different dimensions?
Remark: If $D$ is separable ($Z(D)$ a separable ...

**0**

votes

**2**answers

336 views

### Counting Divisors in $\mathbb{Z}^n$

Basically, I'm looking for ways to multiply elements of $\mathbb{R}^n$ that allow me to count divisors in $\mathbb{Z}^n$.
For every positive integer $n$, I'm looking for an algebra structure on $\...

**10**

votes

**2**answers

810 views

### Are there nonlinear projective spaces?

This is actually a series of questions posed by Guram Berishvili about the structure he calls marao. Everything I am going to write here I took (and messed up) from his home page which is all in ...

**1**

vote

**1**answer

234 views

### cubic forms and finiteness of $k^*/(k^*)^3$

In some recent computation I came across certain cubic forms and was wondering about analogue of following result for quadratic forms.
If $k^*/(k^*)^2$ is finite then there are only finitely many ...

**5**

votes

**1**answer

176 views

### reduced norm from degree 3 division algebra

Let $D$ be a degree $3$ division algebra over a field $k$ of char not 2 and 3.
Any such division algebra is cyclic. I am interested in knowing the cases when the reduced norm map $Nrd : D^* \...

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**0**answers

266 views

### When is a crossed-product algebra a division algebra?

Let $L/K$ be a finite Galois extension with Galois group $G$. For every 2-cocycle $\gamma$ of $G$ with values in $L^\times$ there is the crossed-product $K$-algebra
$$S(L,G,\gamma) = \bigoplus_{g\in ...

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**2**answers

405 views

### Central division algebras and splitting fields

Let $K$ be a field and $D$ be a central division algebra over $K$ of degree $n$. Suppose that $L\subset D$ is a maximal subfield, so that $[L:K]=n$. Then we know that $L$ is a splitting field, so ...

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247 views

### Rational cohomology of the Rosenfeld projective planes

The bioctonionic plane $(\mathbb{C} \otimes \mathbb{O})\mathbb{P}^2$, the quarteroctonionic plane $(\mathbb{H} \otimes \mathbb{O})\mathbb{P}^2$ and the octooctonionic plane $(\mathbb{O} \otimes \...

**1**

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**0**answers

254 views

### Is there any construction of infinite dimensional algebraic division ring? [closed]

I know that there is a division algebra over $\mathbb{Q}$ such that it is algebraic and infinite dimensional over it's center i.e. $\mathbb{Q}$. But for construct this division algebra. we can use ...

**4**

votes

**3**answers

226 views

### 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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**1**answer

422 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**

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**1**answer

540 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**

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**0**answers

380 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 ...

**4**

votes

**1**answer

424 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 ...

**7**

votes

**1**answer

182 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

**1**answer

200 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 ...

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**0**answers

703 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 $\sigma=Int(i)\...

**5**

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**1**answer

266 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

**2**answers

838 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 ...

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**1**answer

827 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

**2**answers

718 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

**1**answer

261 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?