Questions tagged [division-algebras]

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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 ...
John Baez's user avatar
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18 votes
3 answers
981 views

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 ...
Kevin Buzzard's user avatar
17 votes
4 answers
2k views

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 ...
Kevin Buzzard's user avatar
17 votes
1 answer
648 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 $\...
user avatar
15 votes
3 answers
1k views

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 ...
Tim Dokchitser's user avatar
14 votes
1 answer
2k views

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 ...
Adam Epstein's user avatar
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12 votes
1 answer
993 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 ...
GreginGre's user avatar
  • 183
12 votes
1 answer
317 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 $\mathbb{R}$, $\mathbb{C}$, $\...
Maximilian Keßler's user avatar
10 votes
1 answer
591 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 ...
Ben Webster's user avatar
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10 votes
2 answers
874 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 ...
მამუკა ჯიბლაძე's user avatar
9 votes
2 answers
292 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 $...
BPK's user avatar
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9 votes
1 answer
468 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 $u$...
user300's user avatar
  • 215
9 votes
1 answer
510 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 ...
Oliver Braun's user avatar
8 votes
2 answers
275 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(...
sampath's user avatar
  • 255
8 votes
0 answers
295 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 $s_1s_2$ ...
thierry stulemeijer's user avatar
7 votes
1 answer
215 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 ...
Koushik's user avatar
  • 2,076
7 votes
0 answers
207 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 ...
sawdada's user avatar
  • 6,148
7 votes
0 answers
481 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 ...
sampath's user avatar
  • 255
7 votes
1 answer
506 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 ...
Sam Williams's user avatar
6 votes
1 answer
422 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 ...
Vincent's user avatar
  • 2,437
6 votes
1 answer
299 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^* \...
Anupam Singh's user avatar
6 votes
1 answer
340 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 \...
Renee Hoekzema's user avatar
6 votes
1 answer
276 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-...
user15749's user avatar
  • 111
6 votes
0 answers
491 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 ...
jg1896's user avatar
  • 2,683
6 votes
0 answers
296 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 ...
Akiva Weinberger's user avatar
5 votes
3 answers
1k views

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 ...
Mikhail Borovoi's user avatar
5 votes
1 answer
328 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 ...
kneidell's user avatar
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5 votes
1 answer
239 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 $...
Ariel Weiss's user avatar
5 votes
0 answers
161 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 ...
Bob Jones's user avatar
  • 171
4 votes
3 answers
250 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? ...
Māris Ozols's user avatar
4 votes
1 answer
237 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 ...
Lenny's user avatar
  • 43
4 votes
1 answer
111 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 ...
Amir's user avatar
  • 101
4 votes
1 answer
785 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 ...
Hiro's user avatar
  • 945
4 votes
0 answers
220 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 ...
Rita's user avatar
  • 103
4 votes
0 answers
87 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 $k$-...
user237522's user avatar
  • 2,783
4 votes
0 answers
254 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 ...
jacob's user avatar
  • 2,814
4 votes
0 answers
351 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 ...
Steffen Kionke's user avatar
4 votes
0 answers
813 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)\...
user40597's user avatar
3 votes
1 answer
508 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 ...
Emily's user avatar
  • 10.3k
3 votes
1 answer
205 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 ...
Andrei Jaikin's user avatar
3 votes
2 answers
910 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 ...
Tom De Medts's user avatar
  • 6,494
3 votes
1 answer
122 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 ...
Seewoo Lee's user avatar
  • 1,911
3 votes
1 answer
151 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 ...
sampath's user avatar
  • 255
3 votes
1 answer
329 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 ...
Anupam Singh's user avatar
3 votes
1 answer
119 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 ...
Erik D's user avatar
  • 338
3 votes
1 answer
168 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, ...
Anupam Singh's user avatar
3 votes
1 answer
458 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 ...
Soluble's user avatar
  • 1,151
3 votes
0 answers
100 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. ...
Bugs Bunny's user avatar
  • 12.1k
3 votes
0 answers
276 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 ...
Pierre MATSUMI's user avatar
3 votes
0 answers
94 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 $${\...
Mikhail Borovoi's user avatar