The goal of this article is to construct a crossed module representing the cocycle [, ], generating H 3 (g; C) for a simple complex Lie algebra g.

Abstract. We study Lie group structures on groups of the form C ∞ (M,K) , where M is a noncompact 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 ∞ (M,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 Ω 1 (M,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 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 O(M,K) if K is Banach and M is a non-compact complex curve with finitely generated fundamental group.

This paper has two parts. The first part is a review and extension of the methods of integration of Leibniz algebras into Lie racks, including as new feature a new way of integrating 2-cocycles (see Lemma 3.9).In the second part, we use the local integration of a Leibniz algebra h using a Baker-Campbell-Hausdorff type formula in order to deformation quantize its linear dual h * . More precisely, we define a natural rack product on the set of exponential functions which extends to a rack action on C ∞ (h * ).