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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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 ...
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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 $\mathbb{Z}_2-$...
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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 ...
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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^...
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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 ...
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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 ...
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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 ...
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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 \...
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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?
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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 ...
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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 ...
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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 $F$ ...
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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 $F$,...
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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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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 ...
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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 ...
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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 ...
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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],[\...
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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 $A$ ...
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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 ...
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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 ...
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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, ...
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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 ...
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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 ...
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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 $...
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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:
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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 ...
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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 ...
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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 ...
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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 ...
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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 $\Bbb H$...
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Eichler orders in the quaternion.
Suppose $D$ be a quaternion over the number field $K$.
$\Bbb{Definition.}$ The Eichler ordre $R$ of level $N$ in $D$ is defined as the one which satisfies the following equality with the completion $...
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when does a quaternion algebra isomorphic to $M_2(F)$?
We also suppose that the characteristic of a field is not $2.$
Definition 1. An algebra $B$ over $F$ is a quaternion algebra if there exist $i,j\in B$ such that $1,i,j,ij$ is an $F$-basis for $B$ ...
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On well-definedness of Shimura curve.
Suppose that we have a quaternion algebra $D$ over a totally real number field $K$ such that $[K \colon {\Bbb Q}] = {\mathrm{odd}}$. We assume that $D$ splits everywhere at finite places of $K$ and at ...
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Showing there exists no Division algebra on $\mathbb R^3$ indirectly (Sort of...)
Before you read my question please consider that i HAVE to do this exercise as i did below.
So i am showing there exists no division algebra on $\mathbb R^3$.
To show that, i have to show that for $*:...
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Desargues $\implies$ associativity: Projective planes over non-associative structures?
I've been reading about constructing projective planes over division rings (skewfields). There's this very nice fact that if Pappus's theorem holds in a division ring, this ring is actually ...
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An exercise about Division Algebra
In this, page 48, Exercies in chapter 1, there is a following exercise.
Exercise 1. Let $D$ be a division algebra which has finite dimension over the field $k.$ For each $a\in D$ show there is a monic ...
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Splitting fields of subalgebras of central division algebras
Suppose $D$ is a central division algebra over $\mathbb{Q}$ of degree $n$. Let $A \subset D$ be a subalgebra, say also central over $\mathbb{Q}$ (to restrict the degree of generality). Now if a field $...
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A question to Finite Multiplicative subgroups in a division ring of I. N. Herstein
In this, I can't find the results in German as proof steps of Lemma 3 (... by Satz 88 [2, p. 72]) and Theorem 7 in page 123 (... Using results about division subalgebras of division algebras [1, p. 42,...
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"Canonical" norm on a real finite-dimensional unital associative division algebra?
Let $\mathcal{C}$ denote the category of unital associative finite-dimensional division $\mathbb{R}$-algebras. (As a full subcategory of that of unital $\mathbb{R}$-algebras.)
For $A\in \mathcal{C}$ ...
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Prove $\mathbb{Q}$-subalgebra of quaternions is a division algebra over $\mathbb{Q}$
Let D be the $\mathbb{Q}$-subalgebra of $\mathbb{H}$ having basis $1, i, j, k$. Prove that D is a division algebra over $\mathbb{Q}$. This is an exercise from Rotman's Algebra. I only see that is to ...
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Division rings over fields
Given a field $K$. Rings are assumed to be with identity and associative.
Question 1: Is there a (easy) construction of a non-commutative division ring with center $K$?
Question 2: Is there a ...
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Affine domain over an algebraically closed field
An affine domain $A$ over $k$ is a finite dimensional $k$-algebra which is also an integral domain as a ring. Here's my thought.
Fix $a \in A$ and $\varphi:k[x] \rightarrow A $ be defined as $f \...
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Could the discovery of a counterexample to the Unit Conjecture change mathematicians’ understanding of spinors in general?
Recently it has been reported that Giles Gardham has found a counterexample to the Unit Conjecture for group rings, as given in https://arxiv.org/abs/2102.11818 (and also in a popular article https://...
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$\mathbb{R}$ is closed under division
The set of real numbers $\mathbb{R}$ is closed under division. Does that mean $0$ is also considered?
more specifically, should it be $\mathbb{R}-\{ 0 \}$?
because division by $0$ is not defined.
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Uniqueness of the Eichler order
Let ${\mathrm{M}}_2(\widehat{{\cal O}_K})$ be the $2 \times 2$ matrices over the finite adele of the full integer ring ${\cal O}_K$ of a totally real #-field $K$.
For the quaternion algebra $D_K$ over ...
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In a unital division algebra, does the unit have norm 1?
Suppose that $\mathbb{R}^n$ is a unital division algebra.
That is, $\mathbb{R}^n$ is furnished with a bilinear map
$$ *: \mathbb{R}^n\times \mathbb{R}^n\to \mathbb{R}^n $$
such that
$$ x*y = 0 \...
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