On Grassmannians over *-rings
type: Item
instance of: scholarly article
title: On Grassmannians over *-rings
zbMATH Open document ID: 1022.16023
DOI: 10.1016/S0024-3795(02)00344-0
author: Marek Golasiński
author: Francisco Gomez Ruiz
published in: Linear Algebra and its Applications
publication date: 2003-02-05T00:00:00Z
review text: On the ring \(M_n(K)\) of all \(n\times n\) matrices there is an involution * determined by the conjugation and transpose maps, where \(K\) is the field of real numbers \(\mathbb{R}\), the field of complex numbers \(\mathbb{C}\), or the skew-field of quaternions. The tangent bundle \(TG_{n,r}(K)\) of the Grassmannian \(G_{n,r}(K)\) is diffeomorphic to the manifold \(\text{Idem}_{n,r}(K)\) of all idempotent \(n\times n\) matrices with rank \(r\) over \(K\). For any idempotent matrix \(\varphi\) in \(\text{Idem}_{n,r}(K)\) the matrix \(\varphi+\varphi^*-I_n\) is invertible and the matrix \(\psi=\varphi(\varphi+\varphi^*-I_n)^{-1}\) is Hermitian, idempotent with \(\varphi\psi=\psi\) and \(\psi\varphi=\varphi\). Then there exists an explicit regular deformation retraction \(\text{Idem}_{n,r}(K)\to G_{n,r}(K)\) of affine varieties. This paper tends to develop similar results in a wider context by replacing not only \(K\) but even \(M_n(K)\) by any ring with involution. In Section 2 the authors study the matrix ring \(M_n(R)\) over a commutative ring \(R\) with involution. Section 3 is Grassmannians over matrix rings.
Mathematics Subject Classification ID: 16W10
Mathematics Subject Classification ID: 14M15
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 46L05
zbMATH DE Number: 1864739
zbMATH Keywords: tangent bundles
zbMATH Keywords: idempotent matrices
zbMATH Keywords: regular deformation retractions
zbMATH Keywords: affine varieties
zbMATH Keywords: rings with involution
zbMATH Keywords: matrix rings
zbMATH Keywords: Grassmannians
reviewed by: Yueh-er Kuo
MaRDI profile type: Publication
cites work: The regular ring of a finite Baer\(^*\)-ring
cites work: Algebraic characteristic classes for idempotent matrices
cites work: Chern Classes of Projective Modules
description: scientific article; zbMATH DE number 1864739