Normal forms for smooth Poisson structures
type: Item
instance of: scholarly article
title: Normal forms for smooth Poisson structures
zbMATH Open document ID: 0592.58025
DOI: 10.2307/1971210
author: Jack F. Conn
published in: Annals of Mathematics. Second Series
publication date: 1985-01-01T00:00:00Z
review text: In his latest paper [ibid. 119, 577-601 (1984; Zbl 0553.58004)], the author constructed analytic linearizing coordinates for real-analytic Poisson tensors under a certain nondegeneracy condition. In this paper he proves the following linearizing theorem for \(C^{\infty}\) Poisson tensors: Theorem. Let \[ P=()\sum_{1\leq i,j\leq n}P_{ij}(x)\partial /\partial x_ i \wedge \partial /\partial x_ j,\quad with\quad P_{ji}=-P_{ij} \] be a Poisson tensor of class \(C^{\infty}\) defined in a neighborhood of the origin O in \(R^ n\), with \(P(O)=0\). Let \[ P(x)=\sum_{1\leq i<j\leq n,1\leq k\leq n}c^ k_{ij} x_ k \partial /\partial x_ i \wedge \partial /\partial \quad x_ j+ \] \[ \sum_{1\leq i<j\leq n}R_{ij}(x) \partial /\partial x_ i \wedge \partial /\partial x_ j,\quad order(R_{ij})\geq 2 \] be the Taylor expansion to order 1 of P about 0. Suppose that the real Lie algebra \({\mathfrak g}\) for which the scalars \(\{c^ k_{ij}\); \(1\leq i<j\leq n\), \(1\leq k\leq n\}\) form a set of structure constants is semisimple and of compact type. Then P is smoothly linearizable; that is, there exist \(C^{\infty}\) coordinates \(y_ 1,...,y_ n\) defined in a neighborhood of the origin, of the form \(y_ j(x)=x_ j+f_ j(x)\), \(order(f_ j)\geq 2\), in terms of which P is expressed as \[ P(y)=\sum_{1\leq i<j\leq n,1\leq k\leq n}c^ k_{ij} y_ k \partial /\partial y_ i \wedge \partial /\partial \quad y_ j. \] In the proof of this result, the author views the effect upon P of changes of coordinates as a nonlinear partial differential operator. A combination of Newton's method with smoothing operators, as devised by J. Nash and J. Moser, is used to construct successive approximations to the desired coordinate system. The author obtains the following corollary. Let P be a Poisson tensor of class \(C^{\infty}\) on a manifold M. Suppose that \(x\in M\) is a point at which the linear term of the singular part of P is semisimple and of compact type. Then every smooth infinitesimal automorphism of P de…
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 58C15
zbMATH DE Number: 3951635
zbMATH Keywords: Poisson structure
reviewed by: Masahisa Adachi
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/1971210
OpenAlex ID: W2315260204
description: scientific article; zbMATH DE number 3951635