Scaling of Poisson spheres and compact Lie groups
type: Item
instance of: scholarly article
title: Scaling of Poisson spheres and compact Lie groups
zbMATH Open document ID: 1266.53068
published in: The Asian Journal of Mathematics
publication date: 2013-02-27T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.ajm/1355321986
review text: In [Differ. Geom. Appl. 9, No. 1--2, 213--238 (1998; Zbl 0930.37032)], \textit{A. Weinstein} proved that there is no nontrivial smooth scaling \((\phi,\lambda)\) of the standard Bruhat-Poisson structure \(\pi\) on \(\mathrm{SU}(2)\), i.e. a diffeomorphism \(\phi\) of \(\mathrm{SU}(2)\) and a scalar \(\lambda\neq0\) such that \(\phi^*\pi=\lambda^{-1}\pi\), other than \((\phi,\lambda)=(\imath,-1)\) where \(\imath(u):=u^{-1}\) is the inversion map on \(\mathrm{SU}(2)\). This result is generalized to all compact groups with Bruhat-Poisson structure . In the present paper, the author proves that if a scaling \(\phi_{\lambda}\equiv(\phi,\lambda)\) is only required to be continuous on the whole manifold but smooth on each symplectic leaf of a Poisson manifold, then it exists for all \(\lambda>0\) on the Poisson homogeneous space \(S^{2n-1}\) of the Bruhat-Poisson \(\mathrm{SU}(n)\). Also, a Liouville vector field generating \(\phi_\lambda\) is explicitly computed for \(\mathrm{SU}(2)\). Furthermore, if a standard Bruhat-Poisson compact simple Lie group \(K\) is endowed with some stronger topology that is still compatible with the original differential structure on each symplectic leaf of \(K\), then a leafwise smooth and globally continuous scaling \(\phi_\lambda\) exists on \(K\) for all \(\lambda>0\).
reviewed by: Cristian Ida
Mathematics Subject Classification ID: 53D17
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 53D55
zbMATH DE Number: 6139383
zbMATH Keywords: Poisson Lie group
zbMATH Keywords: Bruhat-Poisson structure
zbMATH Keywords: covariant Poisson structure
zbMATH Keywords: homogeneous Poisson structure
zbMATH Keywords: scaling
zbMATH Keywords: deformation quantization
zbMATH Keywords: compact simple Lie group
Wikidata QID: Q115206166
author: Albert Sheu
MaRDI profile type: Publication
OpenAlex ID: W1993016970
DOI: 10.4310/AJM.2012.V16.N4.A8
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description: scientific article; zbMATH DE number 6139383