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^\infty(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,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. We 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.

39 pages

References in corpus (3)

• Lie groups over non-discrete topological fields
• Description of infinite dimensional abelian regular Lie groups
• The Topology of Gauge Groups

Cited by in corpus (16)

• Differentiable mappings on products with different degrees of differentiability in the two factors
• Weighted diffeomorphism groups of Banach spaces and weighted mapping groups
• Fundamentals of submersions and immersions between infinite-dimensional manifolds
• Lie groupoids of mappings taking values in a Lie groupoid
• Obstruction classes of crossed modules of Lie algebroids and Lie groupoids linked to existence of principal bundles
• A Geometric Approach to Noncommutative Principal Torus Bundles
• Free Group Actions from the Viewpoint of Dynamical Systems
• Central extensions of groups of sections
• Complexifications of infinite-dimensional manifolds and new constructions of infinite-dimensional Lie groups
• Slice theorem for Fréchet group actions and covariant symplectic field theory
• Central extensions of Lie groups preserving a differential form
• Principal bundles of embeddings and nonlinear Grassmannians
• Manifolds of mappings on cartesian products
• Exponential laws for spaces of differentiable functions on topological groups
• Spin Structures of Kac-Moody Type
• Holomorphic current groups -- Structure and Orbits