# Questions tagged [division-algebras]

The division-algebras tag has no usage guidance.

84
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### Formulas for the line joining two points in the projective plane over a division algebra

Let $K$ be a[n associative] division algebra (= skew field). By the “projective plane” $\mathbb{P}^2(K)$ over $K$ I mean, as usual, the set of triples $(x,y,z)$ of elements of $K$, not all zero, up ...

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### The number of types of maximal orders in a definite quaternion algebra containing a certain order

I'm referring On the imbeddings of imaginary quadratic orders in definite quaternion orders by Brzezinski and Eichler here.
Let $B$ be a definite quaternion algebra over $\mathbb{Q}$. Given an order $\...

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### Cardinality or covolume of $S$-units in quaternion algebras

Let $D=\left(\frac{a,b}{\mathbb{Q}}\right)$ be a quaternion algebra over $\mathbb{Q}$.
Suppose $D$ is ramified at a finite set $S$ of places and $\infty\in S$.
It is known that the $S$-units (the unit ...

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### Proof of a result by Zhang in Artin's seminal paper

In his seminal paper, Some open problems on three-dimensional graded domains, M. Artin proposed a very small list of possible division rings of fractions that can appear as 'noncommutative function ...

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### Literature for noncommutative birational invariants

Let $k$ be an algebraically closed field of zero characteristic. All fields under discussion are fields over $k$, and all division rings are division algebras over $k$.
There is rich theory of ...

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### Finite dimensional real division algebra up to isotopy

Finite dimensional real division (non necessarily associative) algebras exist in dimensions 1, 2, 4, and 8. The standard example is a Hurwitz algebra $(A,*)$: reals, complexes, quaternions, octonions. ...

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### Moufang identities and Moufang plane

Moufang identities
$$x(y⋅xz)=(xy⋅x)z,$$
$$(zx⋅y)x=z(x⋅yx),$$
$$xy⋅zx=x(yz⋅x)$$
are identities deeply related with alternativity (since setting $z=1$ one recovers left and right alternativity), while a ...

3
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### Modular forms on central division algebra of degree $\ge 3$

I just learned from some online sources including Buzzard's note and Emerton's answer on this MO question about quaternionic modular forms and explicit version of Jacquet-Langlands correspondence in ...

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### What is the quotient group $D^*/{F^*(1+P_D)}$ for a quaternion division algebra $D$ over a local field $F$?

Let $F$ be the non-archimedean local field $\mathbb{Q}_p$ for some prime $p$ and $D$ be a quaternion division algebra over $F$. Let $\mathcal{O}_D$ and $\mathcal{P}_D$ denote the ring of integers of $...

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### Is Hurwitz's theorem true in constructive mathematics?

Hurwitz's theorem says that the only division composition algebras over the real numbers $\mathbb{R}$ are the real numbers themselves $\mathbb{R}$, the complex numbers $\mathbb{C}$, the quaternions $\...

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### Are there any central simple algebras admitting a standard basis?

Are there any central simple algebras admitting a standard basis?
By a standard basis I mean a normal basis that has a cyclic property generalizing that of the familiar basis $1, i, j, k$ for ...

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### Commutator length of the center $Z(D')$ of $D'$ in a division ring $D$

Let $D$ be a division ring, and $D^\times$ the multiplicative group of $D$. Denoted $D'$ (resp. $Z(D')$) by the derived subgroup of $D^\times$ (resp. the center of $D'$). Here, we consider $D'$ ...

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### Product of two involutions in $\mathrm{PSL}_2(D)$

Let $D$ be a division ring and $\mathrm{PSL}_2(D)$. Suppose that $\overline{A}\in\mathrm{PSL}_2(D)$ where $A\in \mathrm{SL}_2(D)$. If $\overline{A}$ is identity, then $\overline{A}$ can express two ...

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### Is there a classification of the $p$-adic normed division algebras?

A normed division algebra over $\mathbb{R}$ is a pair $(A,\lVert{-}\rVert)$ with
$A$ an $\mathbb{R}$-algebra with a unit $1_A$;
$\lVert{-}\rVert\colon A\to\mathbb{R}_{\geq0}$ a norm on $A$;
such ...

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### A problem about extensions of division rings

For a division ring $D$ with center field $F:=Z(D)$ such that $\dim_F D = n^2$, there is a classical result saying that $D\otimes_{F}\bar{F}\cong M_n(\bar{F})$ as $\bar{F}$-algebras, where $\bar{F}$ ...

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### Subalgebra of a crossed product central division algebra, generated by powers of group elements

Let $k=\mathbb C$, let $K$ be a finite extension of the field $k(X_1,X_2,X_3)$ of rational functions in $3$ variables, let $L/K$ be a finite Galois extension of commutative fields and let the Galois ...

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### There is a ring with multiplication. Can we find a formula for division based on formula for multiplication?

Studying divergent integrals, I found a good formula for their multiplication:
$\int_0^\infty f(x)dx\cdot\int_0^\infty g(x)dx=\int_0^\infty D^2 \Delta^{-1} \left(\Delta D^{-2}f(x)\cdot\Delta D^{-2}g(x)...

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### Properties of finite dimensional, real division algebras that yield only $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ and $\mathbb{O}$

It is a classical result by Kervaire and Milnor that every finite-dimensional, real division algebra has dimension 1, 2, 4 or 8, with the most prominent examples being $\mathbb{R}$, $\mathbb{C}$, $\...

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### Eichler orders in a certain quaternion algebra

Let us consider a totally real number field $K$ such that $[K \colon {\Bbb Q}] = {\mathrm{odd}}$. We shall consider the quaternion algebra $D$ over $K$ such that $D$ splits everywhere at finite places ...

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### Multiplication law in a division algebra of dimension 9 over a non-archimedean local field

Let $k$ be a non-archimedean local field, for example, a $p$-adic field (a finite extension of the filed ${\Bbb Q}_p$ of $p$-adic numbers).
It is well known that there is a canonical isomorphism
$${\...

3
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1
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### Charaterisation of quaternion algebras

Let $k$ be a field, and $A$ an associative $k$-algebra with an identity element. Say that $A$ is quadratic if any subalgebra of $A$ generated by a single element has dimension at most two.
I am ...

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### Fixing error in a proof from "Central simple algebras and Galois cohomology"

I'm trying to understand the proof of Proposition 2.2.10 in Gille-Samuely's book "Central simple algebras and Galois cohomology" (2nd edition), and I believe it has an error.
Here's the ...

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286
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### 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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### 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 ...

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### 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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### 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 $u$...

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### 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 ...

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### 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-...

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### 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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### 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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### 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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### 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
$...

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### 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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### 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 ...