Reduced Whitehead groups and the conjugacy problem for special unitary groups of anisotropic Hermitian forms
type: Item
instance of: scholarly article
title: Reduced Whitehead groups and the conjugacy problem for special unitary groups of anisotropic Hermitian forms
zbMATH Open document ID: 1305.19001
published in: Journal of Mathematical Sciences (New York)
publication date: 2014-09-25T00:00:00Z
review text: Let \(K/k\) be a separable field extension of degree 2, \(D\) be a finite-dimensional central division algebra over \(K\) with a \(K/k\)-involution \(\tau,\) \(h\) be an Hermitian anisotropic form on a right \(D\)-vector space with respect to \(\tau\), and let \(\mathrm U(h)\) be the unitary group of \(h\). Then the reduced Whitehead group of its special linear subgroup is defined as follows: \(\mathrm{SU}K_1^{\text{an}} (h)=\mathrm{SU}(h)/[\mathrm U(h),\mathrm U(h)]\), where \([\mathrm U(h),\mathrm U(h)] \) is the commutator subgroup of \(\mathrm U(h)\). The first main result establishes a link between the above group and its analog \(\mathrm{SU}K_ 1 (h) \)for the case of isotropic \(h\) (with respect to the same \(\tau\)). Theorem. There exists a surjective homomorphism \(\mathrm{SU}K_1^{\text{an}} (h)) \to\mathrm{SU}K _1 (h)\). Furthermore, a solution of the conjugacy problem for special unitary subgroups of anisotropic Hermitian forms over quaternion division algebras as subgroups of their multiplicative groups is given.
Mathematics Subject Classification ID: 19B37
Mathematics Subject Classification ID: 16W10
zbMATH DE Number: 6347693
author: V. I. Yanchevskiĭ
reviewed by: L. N. Vaserstein
MaRDI profile type: Publication
full work available at URL: s10958-013-1391-9
OpenAlex ID: W1980947504
cites work: Involutorial simple algebras and real Riemann matrices
cites work: \(R\)-equivalence and special unitary groups
cites work: La $R$-équivalence sur les tores
cites work: \(\text{SK}_1\) of graded division algebras.
cites work: Unitary SK1 of graded and valued division algebras
cites work: Orthogonale Gruppen über algebraischen Zahlkörpern.
cites work: Q4210545
cites work: On finite homomorphic images of the multiplicative group of a division algebra
cites work: Anisotropic groups of type \(A_n\) and the commuting graph of finite simple groups.
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: Q4170852
cites work: Unitary \(\mathrm{SK}_1\) of semiramified graded and valued division algebras.
cites work: Unitary SK\(_1\) for a graded division ring and its quotient division ring.
cites work: On the Commutator Group of a Simple Algebra
cites work: Q4745964
cites work: Q5669427
cites work: Q4106705
cites work: Q4692288
cites work: Q4052349
cites work: Another proof of B. Sury's theorem.
cites work: Q4170904
DOI: 10.1007/S10958-013-1391-9
description: scientific article; zbMATH DE number 6347693