# 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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### 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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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 \... 0 votes 0 answers 41 views ### 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? 4 votes 0 answers 86 views ### 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 ... 1 vote 1 answer 37 views ### 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 ... 3 votes 1 answer 61 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$F$... 5 votes 1 answer 78 views ### 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$F$,... 1 vote 1 answer 178 views ### 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-... 1 vote 1 answer 32 views ### 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 ... 0 votes 0 answers 44 views ### 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 ... 2 votes 1 answer 87 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 ... 3 votes 1 answer 128 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 ... 3 votes 1 answer 78 views ### 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],[\... 1 vote
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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 $A$ ... 1 vote
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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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### 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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### 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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### 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.
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 ...
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 \...