The bisymplectomorphism group of a bounded symmetric domain
type: Item
instance of: scholarly article
title: The bisymplectomorphism group of a bounded symmetric domain
zbMATH Open document ID: 1148.53060
author: Andrea Loi
author: Guy Roos
author: Antonio J. Di Scala
published in: Transformation Groups
publication date: 2008-09-04T00:00:00Z
full work available at URL: https://arxiv.org/abs/0707.2125
review text: Let \(\Omega\) be a Hermitian bounded circled symmetric domain in a complex vector space \(V\). The domain \(\Omega\) is endowed with two natural symplectic forms, the flat form \(\omega_ 0\) and the hyperbolic form \(\omega_ -\). A diffeomorphism \(\varphi : \Omega\to \Omega\) such that \(\varphi^ *\omega_ 0=\omega_ 0\) and \(\varphi^ *\omega_ -=\omega_ -\) is called a bisymplectomorphim. The set of bisymplectomorphims forms an infinite dimensional group. In this paper, the authors determine the group of bisymplectomorphisms of a bounded symmetric domain, which preserve simultaneously the hyperbolic and the flat symplectic form. It is shown that if \(\Omega\) is an irreducible Hermitian bounded circled symmetric domain and if \(K\) is the isotropy group of \(0\), then the analytic or \(C^\infty\) bisymplectomorphisms of \(\Omega\) are the maps \(\varphi=f\circ g\), where \(g=d\varphi(0)\in K\) and \(f\) is associated to \(v(t) = tu(t^ 2)\), with \(u: [0,1)\to S^ 1\simeq U(1)\) analytic or \(C^\infty\) and \(u(0)=1\).
reviewed by: Andrew Bucki
Mathematics Subject Classification ID: 53D05
Mathematics Subject Classification ID: 32M15
zbMATH DE Number: 5320893
zbMATH Keywords: bisymplectomorphim
zbMATH Keywords: Hermitian domain
zbMATH Keywords: symplectic structure
MaRDI profile type: Publication
OpenAlex ID: W1983993599
arXiv ID: 0707.2125
cites work: Symplectic duality of symmetric spaces
cites work: The symplectic structure of Kähler manifolds of non-positive curvature
cites work: Differentiable even functions
DOI: 10.1007/S00031-008-9015-Z
description: scientific article