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Questions tagged [division-algebras]

A division algebra $D$ is a vector spaces over a field $F$ equipped with a bilinear product and a multiplicative neutral element $1$. All the non-zero elements of $D$ have a multiplicative inverse. Associativity is often assumed but not always. Any field is a commutative, associative division algebra. A skewfield = a division ring is always a division algebra over its center. The quaternions form the best known non-commutative division algebra.

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Algebra Isomorphism and embedding between matrix algebra and tensor products of algebras

Let $\mathbb{R}$ denote real numbers. For which $\mathbb{R}$-algebra $A$ are $Mat_n(\mathbb{R} \otimes A)$ and $Mat_n(\mathbb{R}) \otimes A$ algebra isomorphic? If $A$ is finite dimensional with ...
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Wedderburn Artin's theorem for algebras over a field

I learned about the Wedderburn Artin's theorem for simple left artinian ring, says that if $R$ is simple left Artinian ring then $R\cong\mathrm{M}_n(\Delta)$, for some division ring $\Delta$. I want ...
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Group representation classification via Frobenius theorem

I am struggling to understand the following argument on the classification of irreducible representation as real, complex or quaternionic: Let $G$ be a group and $V$ a vector space over $\mathbb{C}$. ...
shamwowexcitante's user avatar
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1 answer
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Sub division rings of dimension 2 of division rings

Suppose $A$ is a division ring and $B$ is a sub division ring such that $A$, as a left vector space over $B$, has dimension $2$. Is it true that $B$ must be commutative ?
Boccherini's user avatar
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Dimension of division ring over a sub division ring

Let $L$ be a division ring ("skew field") and $K$ a sub division ring. Now suppose that $L$, as a left vector space over $K$, has finite dimension $m$. Does $L$, as a right vector space over ...
Boccherini's user avatar
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2 answers
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Is there an Archimedean Dedekind-complete ordered division ring of characteristic zero that is non-isomorphic to the real line?

I know an Archimedean Dedekind-complete ordered field of characteristic $0$ must be isomorphic to $\mathbb{R}$. My question is what if I start from an ordered division ring? Would it still be ...
user760's user avatar
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Can we find a normal nilpotent subgroup $G$ of monomial subgroup of $GL_n(D)$ such that $F[G]=M_n(D)$?

Let $D$ be a non-commutative division ring of finite dimension over its center $F$. Also, $n>1$ is a natural number. Consider that $M$ be the monomial subgroup of $GL_n(D)$(containing $n \times n$ ...
Reza Fallah Moghaddam's user avatar
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Continuous addition and multiplication on Euclidean space (dimension > 2) making it into a field?

While TAing a linear algebra class, I happened upon the following question: Question: for $n\geq 3$, are there continuous operations $+, \cdot : \mathbb R^n \times \mathbb R^n \to \mathbb R^n$ that ...
D.R.'s user avatar
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Is the spectrum of an element in an algebra, $\sigma(x) = \{z \in\mathbb{C}: \vert z\vert\leq\Vert x \Vert\}$? [closed]

In the book, a first course in functional analysis by D.Somasundaram, it is mentioned that $\sigma(x) = \{z \in\mathbb{C}: \vert z\vert\leq\Vert x \Vert\}$ But the proof is given only for one side ...
Easwaran N's user avatar
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Calculating the class number of a maximal order from a Quaternion Algebra

I am currently reading Quaternion Algebras from John Voight and saw how he calculated the class number of a maximal order at example 17.6.3. His quaternion algebra is $B' = \left( \frac{-1,-23}{\...
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An example of local and global containments in quaternion algebras

Let $p$ be a prime. Let $B_{p,\infty}$ be a (unique) quaternion algebra ramified at exactly at $p$ and $\infty$ with a standard basis $1,i,j, k=ij=-ji$. Let $K=\mathbb{Q}(i) \subseteq B_{p,\infty}$ be ...
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A formula for $S$-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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A special case of Skolem-Nother 'sTheorem

Assume that $D$ is a division ring and $n>1$ be a natural number. Let $a\in SL_n(D)$ be a torsion element. For example, $a^m=1$. Also consider that $F=Z(D)$. Therefore, $[F[a]:F]<\infty$. By ...
Reza Fallah Moghaddam's user avatar
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Centre of fixed skew field is the fixed field of centre?

Let $D$ be a skew field, assumed to be finite dimensional over its centre $Z(D)$. Let $\sigma\in\mathrm{Aut} (D)$ be an automorphism of $D$, and let $D^{\sigma}$ be the set of elements in $D$ fixed by ...
Hermetically Sealed Halibut's user avatar
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Isomorphisms between sub division rings of degree $2$

Let $A$ and $B$ be two isomorphic division rings, both contained in the division ring $C$, and suppose that $[C : A] = [C : B] = 2$. Under which assumptions do we know that there exists an ...
Boccherini's user avatar
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3 answers
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Non-associative ring whose non-zero elements form non-commutative quasigroup (with regard to multiplication) without identity?

An example of a non-associative ring R whose non-zero elements form commutative quasigroup (with regard to multiplication in R) without identity is easy to find. I'm looking for examples of a non-...
Peterש's user avatar
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Sub algebras of dimension $2$ of division algebras

Let $U$ be a division algebra over the field $k$, and let $V$ be a sub algebra of dimension $2$. I read (in a recent answer on MathOverflow) that $V$ then necessarily is commutative, that is, $V$ is a ...
Boccherini's user avatar
2 votes
2 answers
94 views

Wedderburn theorem version for superalgebras

I am looking for an example of usage of wedderburn theorem version for superalgebras (which is a $\mathbb{Z}_2-$graded algebra). The theorem states that if $ A$ is finite dimensional $\mathbb{Z}_2-$...
matan's user avatar
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If $x$ in a division $\mathbb{R}$-algebra and $x(a+bi)x^{-1}=a-bi, \forall a,b \in \mathbb{R}$, show that $x^2$ commutes with all $z \in \mathbb{C}$.

If $x$ in a division $\mathbb{R}$-algebra and $x(a+bi)x^{-1}=a-bi, \forall a,b \in \mathbb{R}$, show that $x^2$ commutes with all $z \in \mathbb{C}$. I have tried squaring both sides of $x(a+bi)x^{-1}=...
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The multiplicative group of the real quaternion division ring

Let $\mathbb{H}$ be the real quaternion division ring, that is, $\mathbb{H}$ consists of all elements of the form: $a+bi+cj+dk$ in which $a,b,c,d\in\mathbb{R}$ and $i^2=j^2=k^2=-1,ij=-ji=k$ with usual ...
Tran Nam Son's user avatar
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For quaternions $\alpha$ and $\beta$, find all quaternions $x$ such that $\alpha x=x\beta$.

Let $\mathbb{H}$ be the real quaternion division ring consisting of all elements of form: $a+bi+cj+dk$ in which $a,b,c,d\in\mathbb{R},i^2=j^2=k^2=-1, ij=-ji=k$ and $\alpha=a+bi+cj+dk,\beta=a+\sqrt{b^...
Tran Nam Son's user avatar
2 votes
2 answers
144 views

Division algebras are frobenius algebras

I am following the book Frobenius Algebras I by Andrzej Skowronski and Kunio Yamagata to learn about Frobenius algebras. The goal of Chapter IV, section 5 is to show that finite dimensional semisimple ...
Andarrkor's user avatar
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Did Gelfand prove that every commutative Banach division algebra is either $\mathbb{R}$ or $\mathbb{C}$?

On this MSE page, an answer mentions a cornucopia of vaguely similar results to Hurwitz's Theorem and Frobenius' Theorem, all of which say something like "Every XYZ division algebra is isomorphic ...
Isky Mathews's user avatar
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Let $A$ be a central division algebra (of finite dimension) over a field $k$. Show that $[A,A] \neq A$.

I am looking at the post A central division algebra is not its commutator and I have a few questions regarding the proof that was provided in the answer. Why is $A$ a simple $k$-algebra? My first ...
mathlover314's user avatar
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A division quaternion algebra in which the integral elements don't form a ring

I wish to find a division quaternion algebra $B$ over $\mathbb{Q}$ and elements $\alpha, \beta\in B$ such that $\alpha, \beta$ are integral over $\mathbb{Z}$ but both $\alpha + \beta$ and $\alpha \...
rationalbeing's user avatar
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Are there any infinite-dimesional division algebras over the real numbers? [duplicate]

I have read that the only finite-dimesional division algebras over the real numbers have dimensions 1, 2, 4, and 8. Are there any infinite-dimensional division algebras over the real numbers?
mathlander's user avatar
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Are the dimensions of division algebras over the real numbers related to with generalizations of Euler's four square identity?

I know that the only division algebras over the real numbers have dimension $1, 2, 4,$ and $8$ (real numbers, complex numbers, quaternions, octonions). I also know that those are the only numbers of ...
mathlander's user avatar
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If E is a splitting field for A then it is clear that there exists a finitely generated subfield E'/F that is also a splitting field. [closed]

I saw in Jacobson's book Finite Dimensional Division Algebras (1996) page 158 the sentence "If E is a splitting field for A then it is clear that there exists a finitely generated subfield E'/F ...
Galois1938's user avatar
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1 answer
77 views

division rings $D,D'$ both of finite dimension over their center $F$ are isomorphic as rings iff isomorphic as $F$-algebras?

Say I have two division rings, $D,D'$, both with center $F$ and both are of finite dimension over $F$, for some field $F$. Now suppose that $D\cong D'$ as rings, does if follow that $D\cong D'$ as $F$ ...
mathguy123's user avatar
5 votes
1 answer
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A consequence of Hasse's reciprocity law of simple algebras

I read somewhere in Godement-Jacquet's book (Zeta functions of simple algebras) the following claim: Let $D$ be a division algebra over a number field $F$ of dimension $d^2$. For each place $v$ of $F$,...
youknowwho's user avatar
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1 answer
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Octonions not an associative division algebra?

On this wikipedia page, I read The best-known examples of associative division algebras are the finite-dimensional real ones (that is, algebras over the field R of real numbers, which are finite-...
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1 vote
1 answer
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Tensoring a skew field does not introduce zero divisors

Let $D$ be a skew field with centre $K$ and maximal subfield $E$. Let $F$ be a finite extension of $K$ disjoint to $E$, that is, $F\cap E=K$. Is it true that the tensor product $D\otimes_K F$ has no ...
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Brauer group of a subfield

Let $A$ be a central simple algebra with a finite dimension over the field $F$. Let $A \supset K \supset F$ be a subfield. Show that $C_A(K)$ and $A \otimes_F K$ are both central simple algebras over ...
MZG's user avatar
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2 votes
1 answer
106 views

How to proof there is no idempotent element other than 0 and 1 in a Division Algebra?

If $A$ is a division $K$-algebra. Then I need to proof there is no idempotent element other than $0$ and $1_A$ in $A$. I tried this way : If $0,1_A\neq a\in A$ such that $a^2=a.~$ Now $A$ is division ...
Jayden's user avatar
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1 answer
221 views

Norm of the Sedenions

Let the Cayley-Dickson doubling of the octonions be called the sedenions. The sedenions are not a division algebra, because they contain zero divisors. The presence of zero divisors means that the ...
a196884's user avatar
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1 answer
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Is finite dimensional central division algebra = $\otimes$(proper central division subalgebra)?

One exercise in Jacobson's Basic Algebra II is approximately the following. If $\Delta_1,\Delta_2$ are finite dimensional central division algebra over $F$, and if $\operatorname{gcd}([\Delta_1:F],[\...
user avatar
1 vote
1 answer
141 views

Tensor product of division algebra with $\mathbb R[x]$ or $\mathbb R(x)$

I learned that the tensor product of two division algebras may not be a division algebra. Thus, I am curious if there is some case in which this is true. To be precise, given a division algebra $A$ ...
user avatar
1 vote
1 answer
382 views

Absolutely irreducible/simple $A$-module iff Endomorphism ring consists of scalar matrices

Let $A$ be a non-commutative $K$-algebra (where $K$ a field), whose underlying $K$-vector space is finite dimensional. Definition An $A$-module $M$ is said to be absolutely irreducible or abs. simple ...
user267839's user avatar
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2 votes
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Does every division algebra appear as the endomorphism ring of some group representation? [duplicate]

One way to read Schur's Lemma is The endomorphism ring of an irreducible representation of a finite group over a field $K$ is a division algebra over $K$. For $K=\Bbb R$ there are easy examples ...
M. Winter's user avatar
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Is there any kind of algebraic structure on a line in the hyperbolic plane?

There is a very classical correspondence between projective planes and division algebras: given a plane, each choice of three distinct points (zero, one and infinity) on each line determines addition, ...
მამუკა ჯიბლაძე's user avatar
1 vote
1 answer
173 views

Is there a division algebra of characteristic 2?

Question I need a division algebra D of characteristic 2, of 4 dimensional over its center Z(D), with an element $x\not \in Z(D), x^2\in Z(D)$. Is there any such D? What I know The quaternion algebra ...
Functor's user avatar
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Invertibility conditions on multivectors

Given a split number $n=a+bj$, its inverse is $(a-bj)/(a^2-b^2)=n^*/nn^*$ A similar thing holds for complex and duel numbers. However, it fails for split numbers if $a^2=b^2$ and for dual numbers if $...
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Commutative subalgebras of a matrix algebra over a division algebra

Let $K$ be a field of characteristic $0$, $D$ a central semi-simple division algebra of dimension $d^2$ over $K$, and $n$ a positive integer. Let $R$ be a maximal subfield of $M_n(D)$, then is $\dim_K ...
zom's user avatar
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2 votes
1 answer
135 views

Skew fields with nonzero characteristic

There are many ways to construct a skew field of nonzero characteristic, e.g. the universal field of fractions of a skew polynomial ring $E[x;\sigma]$, a suitable choice of a quaternion algebra over $...
Zerox's user avatar
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2 votes
1 answer
121 views

Relationship between trivial Brauer group and commutative division algebras

Let $k$ be an algebraically closed field and $F$ a finite field extension of the field of rational functions $k(t)$. I've heard two different statements of Tsen's theorem under these conditions: ...
Justin Desrochers's user avatar
2 votes
1 answer
176 views

Maximal subfield of the division algebra is splitting field

Currently I'm trying to get through "An Introduction to Algebraic K-theory" by C.A.Weibel. In the third chapter there is the following example. If $D$ is $d$-dimensional division algebra ...
Jackson Harris's user avatar
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0 answers
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Fields with Finitely Many Division Rings

Let $\mathbb{k}$ be a field. For the purposes of this question, a division ring is a finite-dimensional $\mathbb{k}$-algebra $A$ in which every non-zero object is invertible. Is there a commonly ...
JeCl's user avatar
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1 answer
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The quadratic equation $x^2=c$ in a division ring

Let $D$ be a division ring. We denote $D'$ by the derived subgroup of the multiplicative group $D\setminus\{0\}$, that is, the subgroup generated by all the commutators of $D\setminus\{0\}$. For $c\in ...
Tran Nam Son's user avatar
3 votes
0 answers
170 views

Can I tell from the group how the endomorphism rings of its representations will look like?

Let $G$ be a finite group. It is known that the endomorphism ring of an irreducible real representation $\rho: G\to\mathrm{GL}(\Bbb R^d)$ is (isomorphic to) either $\Bbb R$, $\Bbb C$ or $\Bbb H$ (the ...
M. Winter's user avatar
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1 answer
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Examples of matrix groups with centralizer isomorphic to $\Bbb C$ or $\Bbb H$

By Schur's Lemma, the centralizer $C(G)$ of an irreducible matrix groups $G\subseteq\mathrm{GL}(\Bbb R^d)$ is an $\Bbb R$-division algebra, and thus, isomorphic to either $\Bbb R$, $\Bbb C$ or $\Bbb H$...
M. Winter's user avatar
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