  We construct a generalised notion of Kac-Moody algebras using smooth maps
from the non-compact manifolds ${\cal M}=$SL$(2,\mathbb R)$ and ${\cal M}=$
SL$(2,\mathbb R)/U(1)$ to a finite-dimensional simple Lie group $G$. This
construction is achieved through two equivalent ways: by means of the
Plancherel Theorem and by identifying a Hilbert basis within
$L^2(\mathcal{M})$. We analyse the existence of central extensions and identify
those in duality with Hermitean operators on $\cal M$. By inspecting the
Clebsch-Gordan coefficients of $\mathfrak{sl}(2,\mathbb{R})$, we derive the Lie
brackets characterising the corresponding generalised Kac-Moody algebras. The
root structure of these algebras is identified, and it is shown that an
infinite number of simultaneously commuting operators can be defined.
Furthermore, we briefly touch upon applications of these algebras within the
realm of supergravity, particularly in scenarios where the scalar fields
coordinatize the non-compact manifold $\text{SL}(2,\mathbb{R})/U(1)$.
