# Questions tagged [division-algebras]

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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 ...
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### 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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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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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### 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 ...