An Example in Central Division Algebras

Let D be a central division algebra of degree n over a field K. One defines the genus gen(D) of D as the set of classes [D'] in the Brauer group Br(K) of K represented by central division algebras D' of degree n over K having the same maximal subfields as D. We prove that if the field K is finitely generated and n is prime to its characteristic, then gen(D) is finite, and give explicit estimations of its size in certain situations.

Let D be a central division algebra of degree n over a field K. One defines the genus gen(D) as the set of classes [D′] ∈ Br(K) in the Brauer group of K represented by central division algebras D′ of degree n over K having the same maximal subfields as D. We prove that if the field K is finitely generated and n is prime to its characteristic, then gen(D) is finite, and give explicit estimations of its size in certain situations.

Let $F$ be a global field of positive characteristic, and let ${\\mathcal D}$ be a maximal order in a central division algebra over $F$ . We give a precise properness criterion for moduli spaces of ${\\mathcal D}$ -shtukas with general modifications, which includes the statement that for some division algebras, the space of ordinary ${\\mathcal D}$ -shtukas is not proper. The proof is based on a result on the smoothness of a suitable stack of complete homomorphisms of ${\\mathcal D}$ -modules of rank $1$

For a perfect base field k , we investigate arithmetic aspects of moduli spaces of quiver representations over k : we study actions of the absolute Galois group of k on the k ¯ -valued points of moduli spaces of quiver representations over k and we provide a modular interpretation of the fixed-point set using quiver representations over division algebras, which we reinterpret using moduli spaces of twisted quiver representations (we show that those spaces provide different k -forms of the initial moduli space of quiver representations). Finally, we obtain that stable k ¯ -representations of a quiver are definable over a certain central division algebra over their field of moduli.

Earlier it was proven by Vehkalahti and Lu how the unit group and diversity-multiplexing gain trade-off (DMT) of division algebra-based space-time codes are linked to each other through inverse determinant sums. This work explores this relation further, showing that indeed the density of unit group completely determines the growth of the inverse determinant sum. In particular, in the case of Q(i)-central division algebras, the lower bound obtained from the DMT and the upper bound derived from the growth rate of units coincide.

Let \(D\) be a finite-dimensional central division algebra over a field \(K\). We define the genus \(\mathbf{gen}(D)\) of \(D\) to be the collection of classes \([D^{\prime }] \in \mathrm{Br}(K)\), where \(D^{\prime }\) is a central division \(K\)-algebra having the same maximal subfields as \(D\). In this paper, we describe a general approach to proving the finiteness of \(\mathbf{gen}(D)\) and estimating its size that involves the unramified Brauer group with respect to an appropriate set of discrete valuations of \(K\). This approach is then implemented in some concrete situations, yielding in particular an extension of the Stability Theorem of A. Rapinchuk and I. Rapinchuk (Manuscr. Math. 132:273–293, 2010) from quaternion algebras to arbitrary algebras of exponent two. We also consider an example where the size of the genus can be estimated explicitly. Finally, we offer two generalizations of the genus problem for division algebras: one deals with absolutely almost simple algebraic \(K\)-groups having the same isomorphism/isogeny classes of maximal \(K\)-tori, and the other with the analysis of weakly commensurable Zariski-dense subgroups.

Irreducible strongly indecomposable torsion-free abelian groups of finite rank are called Reid groups, and a full subring R of a division algebra D is called Reid-realizable if it is isomorphic to the endomorphism ring of a Reid group. The main results is that a full subring R of a finite dimensional division algebra D is Reid-realizable if for infinitely many prime numbers p the quotient of the p-adic localizations D p * / div R p * > either is the product of at least two p-adic number fields, or is not a product of division algebras. In particular, a full subring of a finite dimensional central division algebra over an algebraic number field is Reid-realizable if it is locally free for infinitely many prime numbers.

On the genus of a division algebra

Level Zero Types and Hecke Algebras for Local Central Simple Algebras

Let k be a totally real algebraic number field, and let D be a central division algebra of degree d over k.The connected reductive algebraic k-group GL(2, D)/k has k-rank one; it is an inner form of the split k-group GL(2d)/k.We construct automorphic representations π of GL(2d)/k which occur non-trivially in the discrete spectrum of GL(2d, k) and which have specific local components at archimedean as well as non-archimedean places of k so that there exist automorphic representations π of GL(2, D)(A k ) with Ξ(π ) = π under the Jacquet-Langlands correspondence.These requirements depend on the finite set V D of places of k at which D does not split, and on the quest to construct representations π of GL(2, D)(A k ) which either represent cuspidal cohomology classes or give rise to square-integrable classes which are not cuspidal, that is, are eventually represented by a residue of an Eisenstein series.The demand for cohomological relevance gives strong constraints at the archimedean components.

Speh representations over central division algebras

For a central simple algebra A, the group SK 1(A) is defined to be the quotient \(\{a\in A^{\times}\mid \operatorname {\mathit{Nrd}}_{A}(a) = 1\}\big/ [A^{\times},A^{\times}]\), where \(\operatorname {\mathit{Nrd}}_{A}\) is the reduced norm and [A ×,A ×] is the commutator group of A ×. This group SK 1(A) is a subtle invariant of A × that has been of longstanding interest to those working with central simple algebras, algebraic groups, and K-theory. Since SK 1 is a Brauer class invariant, it suffices to compute SK 1 for division algebras. We prove here nearly all the known formulas for SK 1(D) for D a tame central division algebra over a Henselian field. We do this by first obtaining in §11.1 formulas for SK 1 in the graded setting, where calculations are substantially easier and more transparent. We then recover in §11.2 the corresponding formulas in the valued setting by proving in Th. 11.21 that \(\mathit{SK}_{1}(D) \cong \mathit{SK}_{1}(\operatorname {\mathsf {gr}}(D))\) for F=Z(D) Henselian and D tame over F.

Theorem. If a tensor product of a division algebra D D with a quaternion algebra Q Q is not a division algebra, then either D D and Q Q possess a common quadratic subfield or D D contains a splitting field of Q Q not quadratic over the base field. The above theorem generalizes a recently published result of Albert’s. Theorem. If the tensor product of two division algebras over a local or a global field K K is not a division algebra, then they contain a common extension field of K K .

Simple components in group algebras

We study the existence of free subalgebras in division algebras, and prove the following general result: if \(A\) is a noetherian domain which is countably generated over an uncountable algebraically closed field \(k\) of characteristic \(0\), then either the quotient division algebra of \(A\) contains a free algebra on two generators, or it is left algebraic over every maximal subfield. As an application, we prove that if \(k\) is an uncountable algebraically closed field and \(A\) is a finitely generated \(k\)-algebra that is a domain of GK-dimension strictly less than \(3\), then either \(A\) satisfies a polynomial identity, or the quotient division algebra of \(A\) contains a free \(k\)-algebra on two generators.