Mapping groups associated with real-valued function spaces and direct limits of Sobolev-Lie groups
| Autor: | Glockner, Helge, Tarrega, Luis |
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| Rok vydání: | 2022 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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 $\gamma\colon M\to G$ which are $H^s$ 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 $H^s(M,g)$ for $s>m/2$, such that $H^{>m/2}(M,G)$ is the direct limit of the Hilbert-Lie groups $H^s(M,G)$ for $s>m/2$ as a smooth Lie group. 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 $F(M,G)$ whenever real-valued function spaces $F(U,R)$ on open subsets $U$ of $R^m$ are given, subject to simple axioms. Comment: Extended preprint version, 37 pages. Starting point for first author was earlier project "Regularity in Milnor's sense for direct limits of infinite-dimensional Lie groups'' |
| Databáze: | arXiv |
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