  Our goal is to develop a more general scheme for constructing integrable
lattice regularisations of integrable quantum field theories. Considering the
affine Toda theories as examples, we show how to construct such lattice
regularisations using the representation theory of quantum affine algebras.
This requires us to clarify in particular the relations between the light-cone
approach to integrable lattice models and the representation theory of quantum
affine algebras. Both are found to be related in a very natural way, suggesting
a general scheme for the construction of generalised Baxter Q-operators. One of
the main difficulties we need to deal with is coming from the
infinite-dimensionality of the relevant families of representations. It is
handled by means of suitable renormalisation prescriptions defining what may be
called the modular double of quantum affine algebras. This framework allows us
to give a representation-theoretic proof of finite-difference equations
generalising the Baxter equation.
