Questions tagged [division-algebras]
The division-algebras tag has no usage guidance.
81
questions
6
votes
0
answers
479
views
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 ...
- 2,524
1
vote
0
answers
89
views
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 ...
- 2,524
3
votes
0
answers
97
views
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. ...
- 12k
2
votes
3
answers
331
views
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 ...
- 285
3
votes
1
answer
111
views
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 ...
- 1,743
9
votes
2
answers
286
views
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 $...
- 143
17
votes
1
answer
624
views
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 $\...
user173426
0
votes
2
answers
197
views
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 ...
- 103
1
vote
0
answers
53
views
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 $D'$ ...
- 141
2
votes
0
answers
101
views
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 ...
- 141
3
votes
1
answer
481
views
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 ...
- 9,697
2
votes
1
answer
133
views
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 $\bar{F}$ ...
- 321
2
votes
0
answers
48
views
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 ...
- 3,887
1
vote
0
answers
65
views
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)...
- 9,294
12
votes
1
answer
306
views
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 $\mathbb{R}$, $\mathbb{C}$, $\...
3
votes
0
answers
255
views
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 ...
- 751
3
votes
0
answers
93
views
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
$${\...
- 13.4k
3
votes
1
answer
116
views
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 ...
- 338
4
votes
0
answers
215
views
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 ...
- 103
2
votes
1
answer
244
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 ...
- 7,498
3
votes
0
answers
73
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 ...
- 2,524
3
votes
0
answers
202
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 ...
- 2,524
1
vote
1
answer
179
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 ...
- 29
3
votes
0
answers
54
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.
...
- 710
2
votes
1
answer
156
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 ...
- 39
3
votes
1
answer
425
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 ...
- 1,141
3
votes
0
answers
158
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 ...
- 1,378
3
votes
1
answer
199
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 ...
- 1,378
4
votes
1
answer
109
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 ...
- 101
6
votes
0
answers
179
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 ...
- 6,118
0
votes
1
answer
236
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 ...
- 2,829
9
votes
1
answer
459
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$...
- 215
6
votes
0
answers
295
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 ...
- 848
6
votes
1
answer
273
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-...
- 111
3
votes
1
answer
164
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, ...
- 661
3
votes
0
answers
65
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 $\...
- 517
8
votes
0
answers
291
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$ ...
1
vote
0
answers
69
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 ...
-2
votes
1
answer
179
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 ...
- 109
4
votes
0
answers
86
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$-...
- 2,735
5
votes
0
answers
160
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 ...
- 171
6
votes
1
answer
411
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 ...
- 2,427
3
votes
1
answer
144
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 ...
- 255
5
votes
1
answer
234
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
$...
- 824
8
votes
2
answers
264
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(...
- 255
4
votes
0
answers
252
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 ...
- 2,756
7
votes
1
answer
498
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 ...
- 348
2
votes
1
answer
385
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 ...
- 581
0
votes
2
answers
360
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 $\...
- 217
10
votes
2
answers
866
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 ...
- 17.3k