Invariant subalgebras of involutorial quaternion division algebras.
type: Item
instance of: scholarly article
title: Invariant subalgebras of involutorial quaternion division algebras.
zbMATH Open document ID: 1263.16024
author: A. V. Prokopchuk
author: V. I. Yanchevskiĭ
published in: Journal of Mathematical Sciences (New York)
publication date: 2012-11-29T00:00:00Z
review text: The authors continue their study of involutions on algebras and their invariant subalgebras. For a quaternion division algebras \(Q\) over a field \(K\) of characteristic different from \(2\), it is easy to describe the \(K\)-subalgebras of \(Q\) which are invariant under an involution \(\tau\) of the first kind on \(Q\); they are \(Q\), \(K\), \(K(s)\) with \(s\not\in K\), \(s^\tau=\pm s\). When \(\tau\) is an involution of the second kind on \(Q\), with fixed field \(k\), they show the following result (although this may be known in a sense, it is useful to clearly spell it out as the authors have done in this paper): Let \(Q\) be an algebra with a \(K/k\)-involution with \(K\neq k\). For a fixed \(K/k\), the set of quaternion subalgebras of \(Q\) is in bijection with the \(K/k\)-involutions of \(Q\), where an involution \(\tau\) corresponds to a unique quaternion subalgebra \(Q_\tau\) over \(k\) such that \(\tau\) restricts to the canonical involution of \(Q_\tau\). There are also other interesting, useful results proved in the paper.
Mathematics Subject Classification ID: 16K20
Mathematics Subject Classification ID: 16W10
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 11R52
Mathematics Subject Classification ID: 19B99
zbMATH DE Number: 6110979
zbMATH Keywords: quaternion algebras
zbMATH Keywords: involutions of the second kind
zbMATH Keywords: invariant subalgebras
zbMATH Keywords: unitary Whitehead groups
zbMATH Keywords: quaternion division algebras
reviewed by: Balasubramanian Sury
MaRDI profile type: Publication
full work available at URL: s10958-012-0833-0
OpenAlex ID: W2092331256
cites work: Zeros of Successive Derivatives of Entire Functions of the Form h(z)exp(-e z )
cites work: Q4056675
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 \(D\).
DOI: 10.1007/S10958-012-0833-0
description: scientific article; zbMATH DE number 6110979