# Questions tagged [division-algebras]

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### Commutator length of the center $Z(D')$ of $D'$ in a division ring $D$

Let $D$ be a division ring, and $D^\times$ the multiplicative group of $D$. Denoted $D'$ (resp. $Z(D')$) by the derived subgroup of $D^\times$ (resp. the center of $D'$). Here, we consider $D'$ ...
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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 $\mathbb{R}, \mathbb{C}, \mathbb{H}... 0answers 141 views ### Eichler orders in a certain quaternion algebra Let us consider a totally real number field$K$such that$[K \colon {\Bbb Q}] = {\mathrm{odd}}$. We shall consider the quaternion algebra$D$over$K$such that$D$splits everywhere at finite places ... 0answers 71 views ### Multiplication law in a division algebra of dimension 9 over a non-archimedean local field Let$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 $${\... 1answer 101 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 ... 0answers 187 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 ... 1answer 159 views ### The size of endomorphism rings and the relation to ordinariness of Abelian surfaces For Elliptic curves over a finite field, there is a very useful characterization of ordinary elliptic as those with commutative, quadratic endomorphism rings and of supersingular curves as those with ... 0answers 63 views ### On noncommutative transcendence degrees The original transcendence degree for (noncommutative) division algebras is the Gelfand-Kirillov transcendence degree, due to I. Gelfand and K. Kirillov ([ Sur les corps li´es aux algèbres ... 0answers 112 views ### On the Gelfand-Kirillov Conjecture The base field k is of zero characteristic. Notation: A_{n,s}(k):= A_n(k(x_1,\ldots,x_s)), the Weyl agebra over a purely transcedental extension of the base field; F_{n,s}(k), the Weyl field, is ... 1answer 146 views ### Lattices from quaternion algebras (MAGMA software) I am studying the paper "Lattice Packing from Quaternion Algebras" from 2012 about the construction of ideal lattices. In Section 3.3 the authors construct very interesting examples of lattices using ... 0answers 50 views ### A rationality question over skew-fields Let K be a skew-field, k its center, and V a finite-dimensional left vector space over K. Assume that E is a k-subalgebra of \mathrm{End}_K(V), and that E is a (commutative) field. ... 1answer 138 views ### Field of definition of a finite dimensional division algebra and how to reduce it Let F be a field, and E/F an infinite algebraic extension. Let D be a finite dimensional division algebra over E (meaning its center is also E). Is it possible to somehow gow down to a finite ... 1answer 286 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 ... 0answers 115 views ### Free skew fields over sets of different cardinal Let K be a field and let X be a set. Denote by \mathcal D_K(X) the free skew K-field on X. Assume that |X|\ne |Y|. Is it true that \mathcal D_K(X) and \mathcal D_K(Y) are not ... 1answer 153 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 ... 1answer 71 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 ... 0answers 147 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 ... 1answer 192 views ### Classification of finite-dimensional real super C*-algebras The title says it all. I feel like one should be able to find this somewhere, but every time I try to google, I just get results for "super Lie algebras". Does anybody know a reference? I am not so ... 1answer 277 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... 0answers 224 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 ... 1answer 215 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-... 1answer 145 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, ... 0answers 63 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 \... 0answers 263 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 ... 0answers 58 views ### Isotropy of skew-Hermitian forms over division algebras Assume char(F) \neq 2. Let D be a central division algebra over a field F and h: V \rightarrow D be an anisotropic skew-Hermitian form. We can easily see that h_{\bar{F}} is totally ... 1answer 172 views ### When is the following fraction an integer \frac{3^a}{2^b-3}[(\frac{2^b}{3})^c - 1] where a,b,c \in \mathbb{Z}? [closed] I'm trying to evaluate the following fraction \frac{3^a}{2^b-3}[(\frac{2^b}{3})^c - 1], but I'm getting stuck using gcd arguments or divisibility arguments. This is part of an ongoing research I'm ... 0answers 82 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-... 0answers 137 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 ... 1answer 338 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 ... 1answer 110 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 ... 1answer 186 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 ... 2answers 220 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(... 0answers 235 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 ... 1answer 488 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 ... 1answer 316 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 ... 2answers 342 views ### Counting Divisors in \mathbb{Z}^n Basically, I'm looking for ways to multiply elements of \mathbb{R}^n that allow me to count divisors in \mathbb{Z}^n. For every positive integer n, I'm looking for an algebra structure on \... 2answers 829 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 ... 1answer 238 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 ... 1answer 212 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^* \... 0answers 288 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 ... 2answers 471 views ### Central division algebras and splitting fields Let$K$be a field and$D$be a central division algebra over$K$of degree$n$. Suppose that$L\subset D$is a maximal subfield, so that$[L:K]=n$. Then we know that$L$is a splitting field, so ... 1answer 280 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 \...
I know that there is a division algebra over $\mathbb{Q}$ such that it is algebraic and infinite dimensional over it's center i.e. $\mathbb{Q}$. But for construct this division algebra. we can use ...
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? ...