Centralizers of certain matrices relative to the operation related to a \(K\)-loop
type: Item
instance of: scholarly article
title: Centralizers of certain matrices relative to the operation related to a \(K\)-loop
zbMATH Open document ID: 0938.20056
DOI: 10.1007/BF03322103
author: Bokhee Im
author: Helmut Karzel
published in: Results in Mathematics
publication date: 2000-06-20T00:00:00Z
review text: It is known that the point set of the classical three-dimensional hyperbolic geometry can be described algebraically by the set \({\mathbf h}^{1+}\) consisting of all Hermitian \(2\times 2\) matrices having determinant 1 and positive trace. The latter enjoys the property that any \(X\in{\mathbf h}^{1+}\) admits a (unique) square root in \({\mathbf h}^{1+}\) and this implies that the operation \(\oplus\) defined by \(A\oplus B:=(AB^2A)^{{1\over 2}}\) makes \(({\mathbf h}^{1+},\oplus)\) a \(K\)-loop. In this note, the authors adopt a more general point of view upon considering in the vector space \(M_2(K)\) (\(2\times 2\) matrices with entries in a field \(K\)) the operation \(*\) defined by setting \(A*B:=AB^2A\). They end up (cf. Theorems 2.1 and 3.1) with a complete and exhaustive classification of the centralizers \(Z_*(A):=\{X\in M_2(K)\mid A*X=X*A\}\) as \(A\) varies in \(M_2(K)\), viewing the latter as a metric vector space with respect to the quadratic form \(q(X)=\det X\).
reviewed by: Elena Zizioli
Mathematics Subject Classification ID: 20N05
Mathematics Subject Classification ID: 15A30
Mathematics Subject Classification ID: 51A25
zbMATH DE Number: 1346915
zbMATH Keywords: \(K\)-loops
zbMATH Keywords: centralizers
zbMATH Keywords: vector spaces
zbMATH Keywords: quadratic forms
MaRDI profile type: Publication
cites work: \(K\)-loops and quasidirect products in \(2\)-dimensional linear groups over a Pythagorean field
cites work: A geometric construction of the K-loop of a hyperbolic space
Recommended article: Q3318917
Recommended article: On the centralizer of the centralizer of a matrix
Recommended article: Q3054684
Recommended article: Q6052829
Recommended article: Structure of centralizer matrix algebras
Recommended article: σ-centralizers of generalized matrix algebras
Recommended article: Centralizers in free associative algebras and generic matrices
Recommended article: Matrix units for centralizer algebras
Recommended article: THE DIMENSION OF CENTRALISERS OF MATRICES OF ORDER
Recommended article: On the Matrices of Central Linear Mappings
description: scientific article; zbMATH DE number 1346915