A Lie group structure for Fourier integral operators
type: Item
instance of: scholarly article
title: A Lie group structure for Fourier integral operators
zbMATH Open document ID: 0619.58010
DOI: 10.1007/BF01450921
author: R. Schmid
author: Malcolm R. Adams
author: T. S. Ratiu
published in: Mathematische Annalen
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164180
review text: On a compact manifold M, let \({\mathcal D}_{\theta}(T^*M\setminus 0)\) be the group of smooth diffeomorphisms of \(T^*M\setminus 0\) which preserve the canonical one-form \(\theta\). Let \((FIO)_*\) \((resp.(\psi DO)_*)\) denote the group of invertible formal Fourier integral operators (resp. pseudodifferential operators) on M. Then we have an exact sequence of groups \(I\to (\psi DO)_*\to^{j}(FIO)\to^{\pi}{\mathcal D}_{\theta}(T^*M\setminus 0)\to e\), where j is inclusion and \(\pi (A)=\eta\) if graph \(\eta \subset (T^*M\setminus 0)\times (T^*M\setminus 0)\) is the canonical relation associated to A. The authors define a Lie group structure on \((FIO)_*\) by doing this for \((\psi DO)_*\) and \({\mathcal D}_{\theta}(T^*M\setminus 0)\) and constructing a local section of the fibration \(\pi\) satisfying compatibility conditions that allow them to define charts on \((FIO)_*\) that make the group operations \(C^{\infty}\).
Mathematics Subject Classification ID: 58D99
Mathematics Subject Classification ID: 22E65
Mathematics Subject Classification ID: 58J40
zbMATH DE Number: 4005180
zbMATH Keywords: Lie group structure
zbMATH Keywords: Fourier integral operators
zbMATH Keywords: pseudodifferential operators
Wikidata QID: Q115393495
reviewed by: Paul Godin
MaRDI profile type: Publication
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OpenAlex ID: W2028900899
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description: scientific article; zbMATH DE number 4005180