Geometry of \(2 \times 2\) Hermitian matrices over any division ring
type: Item
instance of: scholarly article
title: Geometry of \(2 \times 2\) Hermitian matrices over any division ring
zbMATH Open document ID: 1184.15027
published in: Science in China. Series A
publication date: 2010-01-06T00:00:00Z
review text: Suppose \(D\) is a division ring with an involution \(^{ - }\), \(\mathcal{H}_2 (D)\) is the set of \(2 \times 2\) Hermitian matrices over \(D\). Let \(\text{ad}(A,B) = \text{rank}(A - B)\) be the arithmetic distance between \(A, B \in \mathcal{H}_2 (D)\). The author proves the fundamental theorem of the geometry of \(2 \times 2\) Hermitian matrices over \(D\) \((\text{char}(D) \neq 2)\): If \(\varphi : \mathcal{H}_2 (D) \rightarrow \mathcal{H}_2 (D)\) is the adjacency preserving bijective map, then \(\varphi \) is of the form \(\varphi (X) = {}^t \bar P X^{\sigma} P +\varphi (0)\), where \(P \in GL_{2}(D), \sigma \) is a quasi-automorphism of \(D\). The quasi-automorphism of \(D\) is also investigated.
Mathematics Subject Classification ID: 15B57
Mathematics Subject Classification ID: 15A04
Mathematics Subject Classification ID: 16W20
Mathematics Subject Classification ID: 51D20
Mathematics Subject Classification ID: 16K20
Mathematics Subject Classification ID: 15B33
zbMATH DE Number: 5654016
zbMATH Keywords: division ring
zbMATH Keywords: Hermitian matrix
zbMATH Keywords: quasi-automorphism
zbMATH Keywords: fundamental theorem
author: Liping Huang
MaRDI profile type: Publication
full work available at URL: s11425-009-0070-2
OpenAlex ID: W2266689609
cites work: Q3160258
cites work: Hua's fundamental theorem of the geometry of matrices.
cites work: Geometry of block triangular matrices over a division ring
cites work: Hua's fundamental theorems of the geometry of matrices and related results
cites work: Q4292035
cites work: Q4306605
cites work: Q4835119
cites work: Q4879938
cites work: Geometry of skew-Hermitian matrices
cites work: Geometry of \(2\times 2\) Hermitian matrices
cites work: Geometry of 2×2 Hermitian matrices II
cites work: Adjacency preserving bijection maps of Hermitian matrices over any division ring with an involution
cites work: Adjacency preserving mappings of symmetric and Hermitian matrices
cites work: Q4879942
DOI: 10.1007/S11425-009-0070-2
description: scientific article; zbMATH DE number 5654016