Abstract.
In this paper we study central extensions of the identity component G of the Lie group C ∞(M,K) of smooth maps from a compact manifold M into a Lie group K which might be infinite-dimensional. We restrict our attention to Lie algebra cocycles of the form ω(ξ,η)=[κ(ξ,dη)], where κ:𝔨×𝔨→Y is a symmetric invariant bilinear map on the Lie algebra 𝔨 of K and the values of ω lie in Ω1(M,Y)/dC ∞(M,Y). For such cocycles we show that a corresponding central Lie group extension exists if and only if this is the case for M=𝕊1. If K is finite-dimensional semisimple, this implies the existence of a universal central Lie group extension of G. The groups Diff(M) and C ∞(M,K) act naturally on G by automorphisms. We also show that these smooth actions can be lifted to smooth actions on the central extension if it also is a central extension of the universal covering group G˜ of G.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Author information
Authors and Affiliations
Additional information
Received: 11 April 2002 / Revised version: 28 August 2002 / Published online: 28 March 2003
Rights and permissions
About this article
Cite this article
Maier, P., Neeb, KH. Central extensions of current groups. Math. Ann. 326, 367–415 (2003). https://doi.org/10.1007/s00208-003-0425-x
Issue Date:
DOI: https://doi.org/10.1007/s00208-003-0425-x