Questions tagged [central-simple-algebras]

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8 votes
2 answers
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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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  • 123
0 votes
2 answers
173 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 ...
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  • 103
2 votes
0 answers
43 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 ...
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3 votes
0 answers
97 views

The Brauer group and the second Galois cohomology group

I know that for any finite Galois extension K of $\mathbb{Q}$ the group $H^2(Gal(K/\mathbb{Q}),K^*)$ is isomorphic to the Brauer group $Br(K/\mathbb{Q})$. The isomorphism goes as follows: to a 2-...
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  • 443
3 votes
0 answers
78 views

Multiplication law in a central simple algebra of dimension 9 over a global field

Let $k$ be a global field, for example $k=\Bbb Q$. Let $D$ denote the central simple algebra of dimension 9 over $k$ with given local invariants $i_v$. Here $v$ runs over the set $\Omega_f(k)$ of ...
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2 votes
0 answers
49 views

Equivalence classes of Hermitian elements in a central simple algebra with an involution of second kind

Let $k_0$ be a field of characteristic 0, $k/k_0$ be a quadratic extension, and $A/k$ be a central simple algebra over $k$ of dimension $9=3^2$ with an involution of second kind $\sigma$. Then $^\...
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1 vote
0 answers
48 views

Maximal orders separable over their centre

Let $\mathcal{A}$ be a central simple $K$-algebra, where $K$ is an algebraic number field. It is known that $\mathcal{A}$ is separable over $K$ (following the definition of DeMeyer and Ingraham's book)...
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  • 221
6 votes
1 answer
145 views

Hasse invariant and the Clifford algbera

Let $$q = a_1 x_1^2 + \cdots + a_n x_n^2$$ be a quadratic form over some $p$-adic field $\mathbb{Q}_p$. We thus have its Hasse invariant $$\mathcal{h}(q) = \prod_{1 \leq i < j \leq n} (a_i,a_j)_p \...
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  • 103
4 votes
0 answers
196 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 ...
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  • 103
3 votes
0 answers
78 views

The type number of an algebra

I've been reading On the existence of maximal orders, by C.F. Yu, in which he discusses maximal $R$-orders in semisimple algebras over a field $K$, where $R$ is a Noetherian integral domain and $K = \...
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  • 221
3 votes
0 answers
129 views

Central simple algebras via cohomology

I am following the book Central Simple Algebras and Galois Cohomology, by Gille and Szamuely (I am using the second edition). In section 2.4, the authors remark that the tensor product induces a ...
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3 votes
1 answer
221 views

Elementary classification of division rings

Are there examples (other than the two mentioned below) of fields $K$ such that the classification of all finite dimensional division $K$-algebras is possible using only elementary theory (lets say a ...
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3 votes
0 answers
116 views

Multiplicative structure of the K-theory of Severi-Brauer varieties

There is a well-known result by Quillen stating that if $X_A$ is the Severi-Brauer variety of a central simple algebra $A$ of degree $d$ over a field $k$, then its (Quillen) K-theory decomposes as $$...
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2 votes
0 answers
253 views

Azumaya algebra and central separable algebra over local rings

let be $R$ a commutative ring Auslander and Goldman define a central separabl $R$-algebra $A$ as a faithful $R$-module with $Z(A) = R$ , and such that $A$ is a projective $A\otimes_R A^{\mathrm{op}}$-...
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3 votes
0 answers
59 views

Splitting of central simple algebras in the Schur subgroup over residue fields of places

​Recall that a valuation domain of ​a field extension ​$K/k$ is a $k$-subalgebra $V$ of $K$ ​not equal to $K$ ​such that for every $a\in K$ at least one of $a$ and $a^{-1}$ is in $V$. A​ ​place of $K/...
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6 votes
1 answer
628 views

Example of a central simple algebra

Let $A$ be a finite dimensional central simple algebra over a field $F$ of characteristic $0$. So by Weddernburn's theorem, $A\cong M_n(D)$ for some division algebra $D$ over $F$. Let $\dim_F(D)=m^2$....
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  • 1,111
3 votes
1 answer
269 views

Cyclotomic fields and splitting of central simple algebras

Let $K$ be a cyclotomic field of degree $n$ and $A$ a central simple algebra over $\mathbb{Q}$ of dimension $n^2$. How can one determine whether there is a $\mathbb{Q}$-algebra embedding $K \...
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  • 163
5 votes
0 answers
460 views

Jacobson radical of a tensor product

Let $R$ be a commutative ring and $A_1$, $A_2$ be $R$-algebras. Is there any general mean to compute the Jacobson radical of the tensor product $A_1\otimes_R A_2$ in terms of ${\rm Rad}(A_i )$, $i=1,2$...
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2 votes
2 answers
147 views

Crossed product division algebra

Let $k$ be a field, $D$ be a crossed product division algebra over $k$, namely $D$ has a maximal subfield which is Galois over $k$. Is it possible $D$ contain some other maximal subfield which is non-...
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  • 5,613
4 votes
1 answer
382 views

Elementary proof that a central simple algebra over a field having a maximal subfield is a cyclic algebra

I'm currently reading the book "Central Simple Algebras and Galois Cohomology" written by Philippe Gille and Tamas Szamuely. In the book, I don't understand a computational proof of the theorem that ...
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  • 951
3 votes
0 answers
64 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 $\...
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  • 497
3 votes
1 answer
72 views

Splitting variety for bicyclic algebras

Let $F$ be a field contains a primitive root of unity of order $p$, where $p$ is a prime number. Let $a,b \in F^\times$, then one can look at the cyclic algebra $(a,b)_p \in {_p}Br(F)$ where ${_p}Br(F)...
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  • 103
6 votes
1 answer
297 views

Brauer groups and field extensions

Let $k$ be a field and $\mathrm{Br}(k)$ the Brauer group of $k$. Let $k \subset L$ be a field extension. Let $b \in \mathrm{Br}(k)$ and denote by $b \otimes L \in \mathrm{Br}(L)$ the base-change of $b$...
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3 votes
0 answers
275 views

Interaction between the Brauer group and abelian extensions

If $k$ is a field of characteristic zero and $k$ has no (non-trivial) abelian extensions (e.g. the composite of all solvable extensions of $\mathbb{Q}$), then $\text{Br(k)} = 0$ by the norm residue ...
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  • 169
5 votes
3 answers
495 views

Reference request: correspondence between central simple algebras and quadratic forms

Let $A$ be an algebra over $k$, $\operatorname{tr_A}(x, y):=\operatorname{tr}(m_{xy})$ be a trace form on $A$, and $V_A$ be its restriction on the orthogonal complement to $1$. I wonder why a map $A \...
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  • 1,930
4 votes
0 answers
246 views

Is a central simple algebra necessarily cyclic if it splits after a cyclic Galois extension?

Let $A$ be a central simple algebra of degree $n$ over $k$, $\dim_kA=n^2$, let $K/k$ be a cyclic galois extension of degree $n$. Suppose $A\times_kK\cong M_n(K)$, does this imply that $A$ is a cyclic ...
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  • 5,613
2 votes
1 answer
336 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 ...
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1 vote
1 answer
297 views

Constructing groups of Type E^{66}_{7,1} having non trivial Tits algebra

This can be considered as a continuation of my last useful question: Constructing groups of Type E7 with certain Tits Index It is known that a quadratic form $q$ of dimension $12$, having splitting ...
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  • 1,287
4 votes
0 answers
301 views

Reference for classification of positive involutions

An involution on a finite dimensional algebra $A$ over $\mathbb{Q}$ is an involutive anti-automorphism of $A$. If $\sigma$ is an involution on $A$, we say that $\sigma$ is positive if $\mathrm{Tr}_{A/\...
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  • 4,129
14 votes
1 answer
909 views

Non-commutative Galois theory

Recall that an finite-dimensional algebra $A$ over a field $k$ is central simple iff there is an iso $A \otimes_k A^{op} \cong M_n(k)$ where $A^{op}$ is the opposite ring and $M_n(k)$ is the matrix ...
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  • 1,682
4 votes
0 answers
305 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 ...
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1 vote
0 answers
193 views

Dimension of the center of a subalgebra of a simple algebra

Let $F$ be a field. Let $A$ be a simple (associative unital) $F$-algebra with center reduced to $F$. Let $B$ be a $F$-subalgebra of $A$; assume that $A$ is can be generated as left $B$-module by $n$ ...
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  • 11
3 votes
1 answer
272 views

On conductors, levels and traces on quaternion algebras

I am currently working on level issues in the division central simple algebra case, say $D$ over a local non-archimedean field $F$ (e.g. $\mathbf{Q}_p$). Let say that $\mathcal{O}_D$ and $\mathcal{O}...
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9 votes
1 answer
531 views

Orders in Central Simple Algebras. Applications

It is known that orders in quaternion algebras (over a number field) are used for constructing geometric objects like hyperbolic orbifolds and Shimura curves. Moreover, if one knows embedding ...
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4 votes
1 answer
167 views

Let $A\in\operatorname{M}_n(F)$ be a matrix, how to prove $\bigcap_{X\in C(A)}C(X)=F[A]=\frac{F[x]}{(m_A(x))}$

I asked the following question on Math Stack Exchange, but no people reply. I know MO is more professional and it is for mathematicians to discuss research problems. Maybe this question is unsuitable ...
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  • 243
4 votes
1 answer
146 views

Does every equivalence class in a Brauer-Wall group have a (graded) division algebra?

It is known that each equivalence class in a Brauer group has a division algebra (or, in other words, every central simple algebra is isomorphic to $\mathrm{Mat}(D)$ for some division algebra $D$). Is ...
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  • 305
6 votes
1 answer
227 views

Infinite dimensional simple algebras of finite degree

Let $F$ be a field and let $B$ be an $F$-algebra. The degree of $B$ over $F$ is the smallest positive integer $\deg_F B = d \geq 1$ such that every element of $B$ satisfies a (monic) polynomial of ...
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  • 2,719
4 votes
1 answer
567 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 ...
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  • 925
3 votes
0 answers
304 views

Discussion of specific arithmetic triangle groups?

Arithmetic triangle groups were classified in Takeuchi, Arithmetic triangle groups, J. Math. Soc. Japan Volume 29, Number 1 (1977), 91-106. The (2,3,7) case was discussed in detail in a number of ...
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  • 12.4k
10 votes
0 answers
655 views

What does the tensor product of two central simple algebras correspond to geometrically?

Let $k$ be a field, assumed to have characteristic $0$ for simplicity (though this probably isn't necessary). Let $A$ be a central simple algebra over $k$ of dimension $n^2$. Then the collection of ...
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14 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 ...
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4 votes
2 answers
1k views

Galois theory of endomorphism rings of irreducible representations

Let $k$ be a field which I don't suppose to be algebraically closed. Then, the endomorphism ring of any irreducible representation of a finite group over $k$ is a division ring. What is known about ...
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3 votes
1 answer
311 views

Special subalgebras of central simple algebras

In this question F is a field and all algebras are finite dimensional F algebras. Let X be the set of all F algebras A for which there exist an F algebra B and an F division algebra D such that F is ...
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  • 279
18 votes
3 answers
956 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 ...
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4 votes
0 answers
347 views

"Cholesky decomposition" X=YY* for p-adic matrices?

Let $E/F$ be a quadratic extension of $p$-adic fields. Consider $M_n(E)$ with the unitary (aka 2nd kind) involution $X \mapsto \sigma(X)^{tr}$, where $\sigma(X)$ denotes the entry-wise application of $...
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14 votes
2 answers
956 views

Central simple algebras approach to classfield theory, Merits of.

As noted earlier, I found reading Weil's book "Basic Number Theory" to be a harrowing experience, and I find his writing to be intrinsically hard to understand, though it is perfectly rigorous and ...
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