Central extensions for loop groups of area-preserving diffeomorphisms and their fuzzy sphere limits

We classify central extensions for the loop group LSDiff(S^2) of area-preserving diffeomorphisms of the 2-sphere, and of related twisted loop groups. We then show that the corresponding Lie algebra cocycles are `fuzzy sphere limits’ of Kac-Moody cocycles for (twisted) loop algebras Lsu(k+1) for the limit of large k, provided that the cocycles are rescaled by 6/k^3.