  A generalised notion of Kac-Moody algebra is defined using smooth maps from a
compact real manifold $\mathcal{M}$ to a finite-dimensional Lie group, by means
of complete orthonormal bases for a Hermitian inner product on the manifold and
a Fourier expansion. The Peter--Weyl theorem for the case of manifolds related
to compact Lie groups and coset spaces is discussed, and appropriate Hilbert
bases for the space $L^{2}(\mathcal{M})$ of square-integrable functions are
constructed. It is shown that such bases are characterised by the
representation theory of the compact Lie group, from which a complete set of
labelling operator is obtained. The existence of central extensions of
generalised Kac-Moody algebras is analysed using a duality property of
Hermitian operators on the manifold, and the corresponding root systems are
constructed. Several applications of physically relevant compact groups and
coset spaces are discussed.
