Let $M$ be a compact smooth manifold of dimension $m$ (without boundary) and $G$ be a finite-dimensional Lie group, with Lie algebra $\mathfrak g$. Let $H^{>m/2}(M,G)$ be the group of all mappings $\gamma\colon M\to G$ which are $H^s$ for some $s>\frac{m}{2}$. We show that $H^{>m/2}(M,G)$ can be made a regular Lie group in Milnor's sense, modelled on the Silva space $\smash{H^{>m/2}(M,\mathfrak g):=\lim_{\rightarrow s>m/2}H^s(M,\mathfrak g)}$,
such that $H^{>m/2}(M,G)\; =\;\, \lim\limits_{\rightarrow s>m/2}H^s(M,G)$
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 $\mathcal F(M,G)$ whenever function spaces $\mathcal F(U,\mathbb{R})$ on open subsets $U\subseteq\mathbb{R}^m$ are given, subject to simple axioms.
@article{JOLT_2023_33_1_a11,
author = {H. Gloeckner and L. T\~A{\textexclamdown}rrega},
title = {Mapping {Groups} {Associated} with {Real-Valued} {Function} {Spaces} and {Direct} {Limits} of {Sobolev-Lie} {Groups
}},
journal = {Journal of Lie Theory},
pages = {271--296},
year = {2023},
volume = {33},
number = {1},
doi = {10.5802/jolt.1283},
zbl = {1526.22013},
language = {en},
url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.1283/}
}
TY - JOUR AU - H. Gloeckner AU - L. Tárrega TI - Mapping Groups Associated with Real-Valued Function Spaces and Direct Limits of Sobolev-Lie Groups JO - Journal of Lie Theory PY - 2023 SP - 271 EP - 296 VL - 33 IS - 1 UR - https://jolt.centre-mersenne.org/articles/10.5802/jolt.1283/ DO - 10.5802/jolt.1283 LA - en ID - JOLT_2023_33_1_a11 ER -
%0 Journal Article %A H. Gloeckner %A L. Tárrega %T Mapping Groups Associated with Real-Valued Function Spaces and Direct Limits of Sobolev-Lie Groups %J Journal of Lie Theory %D 2023 %P 271-296 %V 33 %N 1 %U https://jolt.centre-mersenne.org/articles/10.5802/jolt.1283/ %R 10.5802/jolt.1283 %G en %F JOLT_2023_33_1_a11
H. Gloeckner; L. Tárrega. Mapping Groups Associated with Real-Valued Function Spaces and Direct Limits of Sobolev-Lie Groups. Journal of Lie Theory, Volume 33 (2023) no. 1, pp. 271-296. doi: 10.5802/jolt.1283
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