An overview of generalised Kac-Moody algebras on compact real manifolds

A generalised notion of Kac-Moody algebra is defined using smooth maps from a compact real manifold M to finite-dimensional Lie group, by means complete orthonormal bases for Hermitian inner product on the and Fourier expansion. The Peter–Weyl theorem case manifolds related groups coset spaces discussed, appropriate Hilbert space L2(M) square-integrable functions are constructed. It shown that such characterised representation theory which set labelling operator obtained. existence central extensions algebras analysed duality property operators manifold, corresponding root systems Several applications physically relevant discussed.