Lie group structures on groups of smooth and holomorphic maps on non-compact manifolds

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 \(C^\infty(M, {\mathfrak{k}})\) 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 \( \Omega^1(M, {\mathfrak{k}})\) 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 \(\mathcal{O}(M, K)\) 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 \({\mathcal{O}}(M, K)\) if K is Banach and M is a non-compact complex curve with finitely generated fundamental group.