Another proof of B. Sury's theorem.
type: Item
instance of: scholarly article
title: Another proof of B. Sury's theorem.
zbMATH Open document ID: 1189.16017
author: A. V. Prokopchuk
author: V. I. Yanchevskiĭ
published in: Journal of Mathematical Sciences (New York)
publication date: 2010-01-19T00:00:00Z
review text: The structure of normal subgroups of the abstract group \(G(k)\) for an absolutely simple, simply connected algebraic group \(G\) over a global field \(k\), is essentially known excepting the \(k\)-anisotropic groups of outer type \(A_n\). This group can be described as \[ \text{SU}(D,\sigma):=\{d\in D^*:dd^\sigma=1,\;N_{\text{red}}(d)=1\} \] for a division algebra \(D\) with an involution \(\sigma\) of the second kind (that is, \(\sigma\) restricts to the nontrivial automorphism of the center \(K\) of \(D\) -- so, \(K\) is a quadratic extension of \(k\)). As \([\text{U}(D,\sigma),\text{U}(D,\sigma)]\) is a normal subgroup of \(\text{SU}(D,\sigma)\), where \(\text{U}(D,\sigma)=\{d\in D:dd^\sigma=1\}\), the first natural question is whether \(\text{SU}(D,\sigma)=[\text{U}(D,\sigma),\text{U}(D,\sigma)]\) in analogy with the classical result \(\text{SL}(1,D)=[D^*,D^*]\) over global fields. In fact, one might consider more general fields which admit such division algebras with involutions of the second kind and, ask for the determination of the quotient group \(\text{SU}(D,\sigma)/[\text{U}(D,\sigma),\text{U}(D,\sigma)]\). In [Proc. Am. Math. Soc. 134, No. 2, 351-354 (2006; Zbl 1088.16016)], the reviewer and \textit{B. A. Sethuraman} showed that this quotient can be infinite for some function fields over number fields. However, the question remains open for general degree division algebras over global fields. In [Arch. Math. 90, No. 6, 493-500 (2008; Zbl 1225.16007)], the reviewer had computed the quotient when \(D\) is a quaternion division algebra and showed it to be nontrivial in general. In the paper under review, the authors give another proof which could perhaps prove useful in treating more general degrees later. The second author has been an authority in the subject for several decades and has contributed copiously. Thus, it is always desirable to see a contribution by him on this even if it is another proof of a known theorem.
Mathematics Subject Classification ID: 16K20
Mathematics Subject Classification ID: 12E15
Mathematics Subject Classification ID: 11R52
Mathematics Subject Classification ID: 16W10
Mathematics Subject Classification ID: 20G30
zbMATH DE Number: 5660152
zbMATH Keywords: division algebras
zbMATH Keywords: involutions of the second kind
zbMATH Keywords: reduced norms
zbMATH Keywords: special unitary groups
zbMATH Keywords: algebraic function fields
zbMATH Keywords: global fields
reviewed by: Balasubramanian Sury
MaRDI profile type: Publication
full work available at URL: s10958-009-9583-z
OpenAlex ID: W1993051219
cites work: Q4692288
cites work: On the Commutator Group of a Simple Algebra
cites work: Q4100280
cites work: A note on the special unitary group of a division algebra
cites work: On \(\text{SU}(1,D)/[\text{U}(1,D),\text{U}(1,D)]\) for a quaternion division algebra \(D\).
cites work: Q3969027
cites work: On Grunwald-Wang's theorem.
cites work: The Norm 1 Group of -Adic Division Algebra
DOI: 10.1007/S10958-009-9583-Z
description: scientific article; zbMATH DE number 5660152