Revêtements du groupe des quantomorphismes de l'oscillateur harmonique. (Covering spaces for the quantomorphism group of the harmonic oscillator)
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 present survey. For instance, if D denotes the group of diffeomorphisms of \({\mathbb{R}}^ n\) fixing 0, then the universal covering group of \(Diff(X)_ 0\) is diffeomorphic to the subspace in the product space \(D\times Gl(n)_ 0\times {\mathbb{R}}^ n\) of all triples (f,A,x) with \(f'(0)= image\) of A in Gl(n). Thus the homotopy of \(Diff({\mathbb{R}}^ n)\) is carried by the relevant finite-dimensional Lie groups acting transitively. The last two pages of the survey are devoted to an analogous construction for the identity component of the group of diffeological automorphisms of a quantum manifold giving a universal covering of \(Quant({\mathbb{R}}^{2n}\times S^ 1)_ 0\), the homotopy being carried by the relevant symplectic group \(Sp(n)_ 0\).
scientific article; zbMATH DE number 4023595
scientific article