Lie group structures on groups of smooth and holomorphic maps on non-compact manifolds |
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Authors: | Karl-Hermann Neeb Friedrich Wagemann |
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Institution: | 1.Fachbereich Mathematik,Technische Universit?t Darmstadt,Darmstadt,Germany;2.Laboratoire de Mathématiques Jean Leray, Faculté des Sciences et Techniques,Université de Nantes,Nantes cedex 3,France |
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Abstract: | We study Lie group structures on groups of the form C
∞(M, K), where M is a non-compact smooth manifold and K is a, possibly infinite-dimensional, Lie group. First we prove that there is at most one Lie group structure with Lie algebra
for which the evaluation map is smooth. We then prove the existence of such a structure if the universal cover of K is diffeomorphic to a locally convex space and if the image of the left logarithmic derivative in is a smooth submanifold, the latter being the case in particular if M is one-dimensional. We also obtain analogs of these results for the group of holomorphic maps on a complex manifold with values in a complex Lie group K. We further show that there exists a natural Lie group structure on if K is Banach and M is a non-compact complex curve with finitely generated fundamental group.
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Keywords: | Infinite-dimensional Lie group Mapping group Smooth compact open topology Group of holomorphic maps Regular Lie group |
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