A note on the special unitary group of a division algebra

Abstract

Abstract. If D is a division algebra with center a number field K and with an involution of the second kind, it is unknown if the group D)] is trivial. We show that, by contrast, if K is a function field in one variable over a number field, and if D is an algebra with center K and with an involution of the second kind, the group SU (1, D)/[U (1, D), U (1, D)] can be infinite in general. We give an infinite class of examples.

Henselian division algebras and reduced unitary Whitehead groups for outer forms of anisotropic algebraic groups of the type $A_n$

“…2. In [31] Sethuraman and Sury proved that for the special symbol algebras D the group SUK an 1 (D, τ ) is infinite.…”

On reduced anisotropic unitary Whitehead groups of henselian division algebras which are either unramified or tamely ramified with commutative residue algebras

“…Однако если в случае изотропных групп имеется развитая теория, описывающая их свойства, то в анизотропном случае проблема описания приведенных групп Уайтхеда остается, несмотря на более 30-летнюю историю ее существования и пристальное внимание к ней специалистов, мало приступной. К настоящему времени в литературе известны лишь три результата, связанных с конкретными алгебрами D [1][2][3]. Поэтому важным является рассмотрение классов специальных алгебр над гензелевыми полями k как полигона для получения новых гипотез о структуре и свойствах групп SUK 1 an (D, τ).…”

On SU(1,D)/[U(1,D),U(1,D)] for a quaternion division algebra D

“…We shall prove some results for general k and later specialize to global fields to get more complete results applying a theorem of Margulis [2]. In contrast with the case of global fields, it was proved in [7] that, if K is a function field in one variable over a number field, and if D is an algebra with center K and with an involution of the second kind, the group SU (1, D)/[U (1, D), U (1, D)] can be infinite in general; we gave an infinite class of such examples. We believe that the results of this note would be known to experts but it has not been set down in print.…”