Questions tagged [division-algebras]
The division-algebras tag has no usage guidance.
84
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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 ...
18
votes
3
answers
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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 ...
17
votes
4
answers
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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 ...
17
votes
1
answer
648
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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 $\...
15
votes
3
answers
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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 ...
14
votes
1
answer
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Finite dimensional real division algebras
A celebrated theorem of Milnor and Kervaire asserts that any finite dimensional (not necessarily associative, unital) division algebra over the real numbers has dimension 1,2,4 or 8. This result is ...
12
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1
answer
993
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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 ...
12
votes
1
answer
317
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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 $\mathbb{R}$, $\mathbb{C}$, $\...
10
votes
1
answer
591
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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 ...
10
votes
2
answers
874
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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 ...
9
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2
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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 $...
9
votes
1
answer
468
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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 $u$...
9
votes
1
answer
510
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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 ...
8
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2
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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(...
8
votes
0
answers
295
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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 $s_1s_2$ ...
7
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1
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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 ...
7
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0
answers
207
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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 ...
7
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0
answers
481
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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 ...
7
votes
1
answer
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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 ...
6
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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 ...
6
votes
1
answer
299
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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^* \...
6
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1
answer
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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 \...
6
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1
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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-...
6
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0
answers
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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 ...
6
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0
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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 ...
5
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3
answers
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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 ...
5
votes
1
answer
328
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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 ...
5
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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
$...
5
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0
answers
161
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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 ...
4
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3
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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? ...
4
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1
answer
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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 ...
4
votes
1
answer
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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 ...
4
votes
1
answer
785
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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 ...
4
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0
answers
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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 ...
4
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0
answers
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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 $k$-...
4
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0
answers
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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 ...
4
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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 ...
4
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0
answers
813
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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)\...
3
votes
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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 ...
3
votes
1
answer
205
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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 ...
3
votes
2
answers
910
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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 ...
3
votes
1
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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 ...
3
votes
1
answer
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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 ...
3
votes
1
answer
329
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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 ...
3
votes
1
answer
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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 ...
3
votes
1
answer
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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, ...
3
votes
1
answer
458
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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 ...
3
votes
0
answers
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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. ...
3
votes
0
answers
276
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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 ...
3
votes
0
answers
94
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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
$${\...