  We introduce a parametrization of formal deformations of Verma modules of
$\mathfrak{sl}_2$. A point in the moduli space is called a colouring. We prove
that for each colouring $\psi$ satisfying a regularity condition, there is a
formal deformation $U_h(\psi)$ of $U(\mathfrak{sl}_2)$ acting on the deformed
Verma modules. We retrieve in particular the quantum algebra
$U_h(\mathfrak{sl}_2)$ from a colouring by $q$-numbers. More generally, we
establish that regular colourings parametrize a broad family of formal
deformations of the Chevalley-Serre presentation of $U(\mathfrak{sl}_2)$. The
present paper is the first of a series aimed to lay the foundations of a new
approach to deformations of Kac-Moody algebras and of their representations. We
will employ in a forthcoming paper coloured Kac-Moody algebras to give a
positive answer to E. Frenkel and D. Hernandez's conjectures on Langlands
duality in quantum group theory.
