On linearized Poisson structures.
type: Item
instance of: scholarly article
title: On linearized Poisson structures.
zbMATH Open document ID: 1039.53093
DOI: 10.1023/A:1012324618578
author: Yurii Vorobiev
published in: Mathematical Notes
publication date: 2003-06-15T00:00:00Z
review text: The linearizability of a given Poisson structure \(\Psi \) at a non-zero dimensional symplectic leaf B (\(\dim B\geq 1\)) is studied. Two basic structures are introduced: a normal bundle \(E\) to the symplectic leaf \(B\) and the pull-back \(\Psi _{f}\) of the Poisson structure \(\Psi \) to \(E\) via an exponential map \(f\). Using the idea of minimal coupling, it is shown that there exists a linearized Poisson structure \(\Pi _{L}\) of \(B\) that gives a first approximation to \(\Psi _{f}\). This linearized structure \(\Pi _{L}\) is uniquely determined by the choice of a subbundle \(L\) transversal to \(B\). The main result of the paper is that in a neighborhood of the zero section \(B\) in \(E\), the linearized Poisson structure \(\Pi _{L}\) is independent of the choice of a transversal \(L\) up to a diffeomorphism identical on \(B\). It is concluded that this result leads to the natural setting of the linearization problem for Poisson structures near embedded symplectic leaves.
Mathematics Subject Classification ID: 53D17
Mathematics Subject Classification ID: 37J05
zbMATH DE Number: 1927778
zbMATH Keywords: linearized Poisson structure
zbMATH Keywords: symplectic leaf
zbMATH Keywords: minimal coupling
zbMATH Keywords: transversal subbundle
reviewed by: Gheorghe Zet
MaRDI profile type: Publication
full work available at URL: a:1012324618578
OpenAlex ID: W45979754
description: scientific article; zbMATH DE number 1927778