Cayley map for symplectic groups
type: Item
instance of: scholarly article
title: Cayley map for symplectic groups
zbMATH Open document ID: 1484.22016
publication date: 2021-09-03T00:00:00Z
review text: A \(2n\!\times\!2n\) matrix \(g\) belongs to the symplectic group \(\mathrm{Sp}(2n,\mathbb{R}\)) if and only if \(g^tJg=J\), and a \(2n\!\times\!2n\) matrix \(X\) belongs to the corresponding Lie algebra \(sp(2n,\mathbb{R})\) if and only if \(X^tJ+JX=0\). The matrix \(J\equiv J_{2n}\) satisfying \(J^2=-I\) and \(J^t=-J\) is the symplectic unit. Since \(X^tJ+JX=0\ \Rightarrow \ (JX)^t=JX\), the Lie algebra of the symplectic group can be described by using the symmetric matrices as \(sp(2n,\mathbb{R})=\{\, JS\ |\ S^t=S\, \}\). By using this description and the Cayley map \(X\mapsto (I+X)/(I-X)\), the authors obtain an interesting parametrization for the elements of \(\mathrm{Sp}(2n,\mathbb{R}\)). They describe in terms of parameters the inverse of an element and the composition of two elements. Some numerical examples are also included. For the entire collection see [Zbl 1468.53002].
reviewed by: Nicolae Cotfas
Mathematics Subject Classification ID: 22E70
Mathematics Subject Classification ID: 81R05
Mathematics Subject Classification ID: 17B81
zbMATH DE Number: 7391058
zbMATH Keywords: Cayley formula
zbMATH Keywords: Cayley map
zbMATH Keywords: group composition
zbMATH Keywords: Hamiltonian matrices
zbMATH Keywords: Lie algebra
zbMATH Keywords: Lie group
zbMATH Keywords: rotations
zbMATH Keywords: symmetric matrices
zbMATH Keywords: symplectic matrices
zbMATH Keywords: vector parameterization
author: Clementina Mladenova
author: Ivaïlo Mladenov
MaRDI profile type: Publication
full work available at URL: giq-22-2021-154-164
OpenAlex ID: W3161556976
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DOI: 10.7546/GIQ-22-2021-154-164
description: scientific article; zbMATH DE number 7391058