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 γ of G with values in L× there is the crossed-product K-algebra S(L,G,γ)=⨁g∈GLeg with the multiplication (μeg)⋅(λeh)=μg(λ)γ(g,h)egh for all λ,μ∈L. The K-algebra S(L,G,γ) is a central simple K-algebra. In fact, the isomorphism class of S(L,G,γ) only depends on the cohomology class … Read more

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 non-empty, and if so what is the dimension of this space? I always thought this was a … Read more

Is x∈A1x \in A_1 left algebraic over the subalgebra generated by pp and qq, [q,p]=1[q,p]=1?

Let A1:=A1(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)↦(p,q) be a k-algebra homomorphism of A1, so [q,p]=1. Denote the image of A1 under f by T (T=A1(p,q,k) is the k-subalgebra of A1 generated by p and q). … Read more

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 setup (which I’ll try to write in such a way that you won’t need the book to follow this). Let k be an infinite field and … Read more

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 called elliptic if it is semisimple and the maximal split subtorus of the center of the centralizer … Read more

Is an associative division algebra required for this phenomenon?

For which integers d≥1 can we find real matrices R1,…,Rd of size d×d such that for any unit vector v∈Rd, R1v,…,Rdv is an orthonormal basis? Note that the chosen set of Ris has to work simultaneously for all v. (Does this phenomenon have a name?) It may not be obvious at first, but this question … Read more

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 char p>0 and let R=End(C), which is a maximal order in a quaternion algebra over Q. Then he uses the fact that any projective module of rank g≥2 over R is … Read more

Properties of finite dimensional, real division algebras that only yield R,C,H\mathbb{R}, \mathbb{C}, \mathbb{H} and O\mathbb{O}

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 R,C,H and O. However, as far as I am aware, a full classification of all such finite-dimensional division algebras is still unknown, although there are plenty of classification results … Read more

Is SL1(D)SL_1(D) toplogically finitely generated, for DD a division algebra over a local field?

I’ve been struggling with this one all day, and I was wondering if someone can give me a hand with the proof. I’m not even sure if the group in question is finitely generated, so I would appreciate if anyone will tell me otherwise as well. remark 1 I’ve posted this question on Math.stackexchange as … Read more