On maximal subgroups of the multiplicative group of a division algebra

This note studies the question whether a multiplicative group of a division ring has a maximal subgroup. It is published in J. Algebra. This is a reputable journal in the subject algebra. Most of submitted papers from 5* schools in RAE was in this journal.

SK1-like functors for division algebras

We investigate the group valued functor G(D) = D*/F*D' where D is a division algebra with center F and D' the commutator subgroup of D*. We show that G has the most important functorial properties of the reduced Whitehead group SK1. We then establish a fundamental connection between this group, its residue version, and relative value group when D is a Henselian division algebra. The structure of G(D) turns out to carry significant information about the arithmetic of D. Along these lines, we employ G(D) to compute the group SK1(D). As an application, we obtain theorems of reduced K-theory which require heavy machinery, as simple examples of our method.

SK1 of graded division algebras

The reduced Whitehead group $\SK$ of a graded division algebra graded by a torsion-free abelian group is studied. It is observed that the computations here are much more straightforward than in the non-graded setting. Bridges to the ungraded case are then established by the following two theorems: It is proved that $\SK$ of a tame valued division algebra over a henselian field coincides with $\SK$ of its associated graded division algebra. Furthermore, it is shown that $\SK$ of a graded division algebra is isomorphic to $\SK$ of its quotient division algebra. The first theorem gives the established formulas for the reduced Whitehead group of certain valued division algebras in a unified manner, whereas the latter theorem covers the stability of reduced Whitehead groups, and also describes $\SK$ for generic abelian crossed products.

Unitary SK1 of unitary and graded division algebras

The reduced unitary Whitehead group $\SK$ of a graded division algebra equipped with a unitary involution (i.e., an involution of the second kind) and graded by a torsion-free abelian group is studied. It is shown that calculations in the graded setting are much simpler than their nongraded counterparts. The bridge to the non-graded case is established by proving that the unitary $\SK$ of a tame valued division algebra wih a unitary involution over a henselian field coincides with the unitary $\SK$ of its associated graded division algebra. As a consequence, the graded approach allows us not only to recover results available in the literature with substantially easier proofs, but also to calculate the unitary $\SK$ for much wider classes of division algebras over henselian fields.

Reduced K-theory for Azumaya algebras

Abstract In the theory of central simple algebras, often we are dealing with abelian groups which arise from the kernel or co-kernel of functors which respect transfer maps (for example K-functors). Since a central simple algebra splits and the functors above are “trivial” in the split case, one can prove certain calculus on these functors. The common examples are kernel or co-kernel of the maps Ki(F)?Ki(D), where Ki are Quillen K-groups, D is a division algebra and F its center, or the homotopy fiber arising from the long exact sequence of above map, or the reduced Whitehead group SK1. In this note we introduce an abstract functor over the category of Azumaya algebras which covers all the functors mentioned above and prove the usual calculus for it. This, for example, immediately shows that K-theory of an Azumaya algebra over a local ring is “almost” the same as K-theory of the base ring. The main result is to prove that reduced K-theory of an Azumaya algebra over a Henselian ring coincides with reduced K-theory of its residue central simple algebra. The note ends with some calculation trying to determine the homotopy fibers mentioned above.