Affine realizations of sphere algebras

The group Map(M,G) of smooth mappings from a compact manifold M to a simple Lie group G is an infinite-dimensional Lie group. The simplest case is when M=S-1. The central extension of the corresponding Lie algebra Map(S-1,g) (with g the Lie algebra of G) is an affine Kac-Moody algebra. The representations of Map(S-1,g) and of its central extension have been thoroughly investigated. Less work has been done for higher-dimensional M. We discuss the particular cases when M is the two- and three-spheres S-n, n=2,3. In this work we construct particular realizations of the centrally extended Map(S-n,g) from a given realization of an affine Kac-Moody algebra. (C) 1997 American Institute of Physics.