  We propose in this thesis a new deformation process of Kac-Moody algebras and
their representations. The direction of deformation is given by a collection of
numbers, called a colouring. The natural numbers lead for example to the
classical algebras, while the quantum numbers lead to the associated quantum
algebras. We first establish sufficient and necessary conditions on colourings
to allow the process depend polynomially on a formal parameter and to provide
the generalised quantum enveloping (GQE) algebras. We then lift the
restrictions and show that the process still exists via the coloured Kac Moody
algebras. We formulate the GQE conjecture which predicts that every
representation in the category Oint of a Kac-Moody algebra can be deformed into
a representation of an associated GQE algebra. We give various evidences for
this conjecture and make a first step towards its resolution by proving that
Kac-Moody algebras without Serre relations can be deformed into GQE algebras
without Serre relations. In case the conjecture holds, we establish an analog
result for coloured Kac-Moody algebras, we prove that the deformed
representation theories are parallel to the classical one, we explicit a
deformed Serre presentation for GQE algebras, we prove that the latter are the
representatives of a natural class of formal deformations of Kac-Moody algebras
and are h-trivial in finite type. As an application, we explain in terms of
interpolation both classical and quantum Langlands dualities between
representations of Lie algebras, and we propose a new approach which aims at
proving a conjecture of Frenkel-Hernandez. In general, we prove that
representations of two isogenic coloured Kac-Moody algebras can be interpolated
by representations of a third one. Observing that standard quantum algebras
satisfy the GQE conjecture, we give a new proof of the previously mentioned
classical Langlands duality.
