2022/10/03 by Helge Glöckner, Glockner, Helge, Luis Tárrega +1 · 1 citation
Mathematics · #22E65 (Primary) 22E67 #46A13 #46E35 #46M40 (Secondary) #Advanced Mathematical Physics Problems #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Group Theory (math.GR)
paper · pdf · doi:10.48550/arxiv.2210.01246
openalex publication_date 2022/10/03 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28
Let M be a compact smooth manifold of dimension m (without boundary) and G be a finite-dimensional Lie group, with Lie algebra g. Let H>m/2(M,G) be the group of all mappings γ\colon M→ G which are Hs for some s>m/2. We show that H>m/2(M,G) can be made a regular Lie group in Milnor's sense, modelled on the Silva space H>m/2(M,g) which is the locally convex direct limit of the Hilbert spaces Hs(M,g) for s>m/2, such that H>m/2(M,G) is the direct limit of the Hilbert-Lie groups Hs(M,G) for s>m/2 as a smooth Lie group. We also explain how the (known) Lie group structure on Hs(M,G) can be obtained as a special case of a general construction of Lie groups F(M,G) whenever real-valued function spaces F(U,R) on open subsets U of Rm are given, subject to simple axioms.