Central extensions of current groups

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.