Journal of Lie Theory 33 (2023), No. 1, 271--296
Copyright Heldermann Verlag 2023
Mapping Groups Associated with Real-Valued Function Spaces and Direct Limits of Sobolev-Lie Groups
Helge Gloeckner
Institut für Mathematik, Universität Paderborn, Germany
glockner@math.upb.de
Luis Tárrega
Departamento de Matemáticas, Universitat Jaume I, Castellón, Spain
ltarrega@uji.es
[Abstract-pdf]
\newcommand{\cg}{{\mathfrak g}} \newcommand{\cF}{{\mathcal F}} \newcommand{\dl}{{\displaystyle \lim_{\longrightarrow}\, }} \newcommand{\sub}{\subseteq} Let $M$ be a compact smooth manifold of dimension $m$ (without boundary) and $G$ be a finite-dimensional Lie group, with Lie algebra $\cg$. Let $H^{>m/2}(M,G)$ be the group of all mappings $\gamma\colon M\to G$ which are $H^s$ for some $s>\frac{m}{2}$. We show that $H^{>m/2}(M,G)$ can be made a regular Lie group in Milnor's sense, modelled on the Silva space $\smash{H^{>m/2}(M,\cg):={\dl}_{s>m/2}H^s(M,\cg)}$, such that \[ H^{>m/2}(M,G)\; =\;\, {\dl}_{s>m/2}H^s(M,G) \] \vskip-5mm as a Lie group (where $H^s(M,G)$ is the Hilbert-Lie group of all $G$-valued $H^s$-mappings on $M$). We also explain how the (known) Lie group structure on $H^s(M,G)$ can be obtained as a special case of a general construction of Lie groups $\cF(M,G)$ whenever function spaces $\cF(U,\mathbb{R})$ on open subsets $U\sub\mathbb{R}^m$ are given, subject to simple axioms.
Keywords: Sobolev space, Banach space-valued section functor, mapping group, direct limit, pushforward, superposition operator, Nemytskij operator.
MSC: 22E65; 22E67, 46A13, 46E35, 46M40.
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