Lie groups of \(C^k\)-maps on non-compact manifolds and the fundamental theorem for Lie group-valued mappings. (English) Zbl 1496.22012
The article studies the existence of Lie group structures on groups of the form \(C^k(M,K), k \in \mathbb{N}\), where \(M\) is a non-compact smooth manifold (possibly with boundary) and \(K\) is a (possibly infinite-dimensional) Lie group. For \(k=\infty\) these groups are known in the literature as current groups. Here Lie group means locally convex Lie group (a la [K.-H. Neeb, Jpn. J. Math. (3) 1, No. 2, 291–468 (2006; Zbl 1161.22012)]), where differentiability is understood in the Bastiani calculus. As a tool, a version of the fundamental theorem for Lie algebra-valued functions is provided. Under some technical conditions (involving the regularity of the target Lie group), the author establishes the existence of Lie group structures on \(C^k(M,K)\). In particular, it is shown that \(C^k (\mathbb{R},K)\) admits a Lie group structure under some conditions on \(K\). These results were a stepping stone for the generalised versions on Lie group structures constructed later in [H. Glöckner and A. Schmeding, Ann. Global Anal. Geom. 61, No. 2, 359–398 (2022; Zbl 1484.58005)].
Reviewer: Alexander Schmeding (Bodø)
MSC:
| 22E65 | Infinite-dimensional Lie groups and their Lie algebras: general properties |
| 46T05 | Infinite-dimensional manifolds |
Keywords:
current group; fundamental theorem for Lie algebra valued functions; functions of mixed order differentiability; compact open topology; locally convex Lie groupReferences:
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