Lie group structures on groups of smooth and holomorphic maps on non-compact manifolds. (English) Zbl 1143.22016

The authors study Lie group structures on groups of the form \(C^{\infty}(M,K)\), where \(M\) is a non-compact smooth manifold and \(K\) is a possibly infinite-dimensional Lie group. The main results of the paper are the following: Theorem A. Let \(K\) be a connected regular real Lie group and \(M\) a real finite-dimensional connected manifold. Then the group \(C^{\infty}(M,K)\) carries a Lie group structure compatible with evaluations in the following cases: (1) The universal covering of \(K\) is diffeomorphic to a locally convex space. If, in addition, \(\pi_1(M)\) is finitely generated, then the Lie group structure is compatible with the smooth compact open topology. (2) \(M\) has dimension one. (3) \(M\equiv \mathbb{R}^k\times C\), where \(C\) is compact. Theorem B. Let \(K\) be a regular complex Lie group and \(M\) a finite-dimensional connected complex manifold without boundary. Then the group of all holomorphic maps on \(M\) with values in \(K\) carries a Lie group structure with Lie algebra compatible with evaluations in the following cases: (1) The universal covering of \(K\) is diffeomorphic to a locally convex space. If, in addition, \(\pi_1(M)\) is finitely generated, then the Lie group structure is compatible with the compact open topology. (2) \(M\) has complex dimension one, \(\pi_1(M)\) is finitely generated, and \(K\) is a Banach-Lie group.


22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
22E67 Loop groups and related constructions, group-theoretic treatment
22E15 General properties and structure of real Lie groups
22E30 Analysis on real and complex Lie groups
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