Revêtements du groupe des quantomorphismes de l'oscillateur harmonique. (Covering spaces for the quantomorphism group of the harmonic oscillator)
type: Item
instance of: scholarly article
title: Revêtements du groupe des quantomorphismes de l'oscillateur harmonique. (Covering spaces for the quantomorphism group of the harmonic oscillator)
zbMATH Open document ID: 0629.22012
DOI: 10.1016/0393-0440(85)90020-8
author: Paul Donato
published in: Journal of Geometry and Physics
publication date: 1985-01-01T00:00:00Z
review text: A diffeology on a set X is a class of test maps from open sets of some Euclidean space into X which contains constants and allows patching and composition. This is specified through three axioms which are satisfied by the class of all \(C^{\infty}\)-maps from open Euclidean domains into X if X itself is a Euclidean domain. A diffeological space is a set equipped with a diffeology. A map \(f: X\to Y\) between diffeological spaces is called differentiable if for each test map p into X the composition f p is a test map into Y. Diffeological spaces and differentiable maps form a category \({\mathcal D}\) which contains the category of differentiable manifolds and differentiable maps as a full subcategory. If all test maps are locally constant on X, then X is a discrete diffeological space. If X is a topological space and the class of test maps consists of all continuous functions of open Euclidean domains into X one obtains a so-called topo-diffeological space. The concepts of path spaces, loop spaces, fundamental group, covering spaces, universal covering space, simple connectivity etc. work in this category in close analogy to what happens in the category of topological spaces and continuous maps. (But one is to be warned about such phenomena as the discreteness of the space \({\mathbb{Q}}\) of rationals in the space of real \({\mathbb{R}}\) with its natural diffeology so that \({\mathbb{R}}\to {\mathbb{R}}/ {\mathbb{Q}}\) becomes a covering map.) The main objective here is to study group objects in the category \({\mathcal D}\). These are called diffeological groups and generalize finite- dimensional Lie groups. In particular, for each differentiable manifold X (or even each diffeological space X), the group Diff(X) of all diffeomorphisms of X carries a unique finest group diffeology making evaluation jointly differentiable. The study of such groups and the explicit construction of their universal covering group in the category \({\mathcal D}\) is the purpose of the …
reviewed by: Karl H. Hofmann
Mathematics Subject Classification ID: 22E65
Mathematics Subject Classification ID: 58B25
Mathematics Subject Classification ID: 22E70
Mathematics Subject Classification ID: 57T20
Mathematics Subject Classification ID: 57R50
zbMATH DE Number: 4023595
zbMATH Keywords: diffeology
zbMATH Keywords: diffeological spaces
zbMATH Keywords: topo-diffeological space
zbMATH Keywords: group objects
zbMATH Keywords: Lie groups
zbMATH Keywords: group of diffeomorphisms
zbMATH Keywords: universal covering group
zbMATH Keywords: quantum manifold
zbMATH Keywords: symplectic group
MaRDI profile type: Publication
full work available at URL: 0393-0440(85)90020-8
OpenAlex ID: W2003693252
cites work: The non-finite homotopy type of some diffeomorphism groups
cites work: Q4166257
cites work: Sur les difféomorphismes de la sphère de dimension trois \(\Gamma_ 4 = 0\)
cites work: An analogue of the Thom isomorphism for crossed products of a C* algebra by an action of R
cites work: Q5610453
cites work: A fibre bundle description of Teichmüller theory
cites work: Q5622916
cites work: Q4046802
cites work: Q3261851
cites work: An axiomatization of the homotopy groups
cites work: Commutators of diffeomorphisms. II
cites work: C*-algebras associated with irrational rotations
cites work: Diffeomorphisms of the 2-Sphere
cites work: Q5580943
cites work: Q4157710
cites work: Q3966795
description: scientific article; zbMATH DE number 4023595