  Quantum groups play a role of symmetries of integrable theories in two
dimensions. They may be detected on the classical level as Poisson-Lie
symmetries of the corresponding phase spaces. We discuss specifically the
Wess-Zumino-Witten conformally invariant quantum field model combining two
chiral parts which describe the left- and right-moving degrees of freedom. On
one hand side, the quantum group plays the role of the symmetry of the chiral
components of the theory. On the other hand, the model admits a lattice
regularization (in the Minkowski space) in which the current algebra symmetry
of the theory also becomes quantum, providing the simplest example of a quantum
group symmetry coupling space-time and internal degrees of freedom. We develop
a free field approach to the representation theory of the lattice $sl(2)$-based
current algebra and show how to use it to rigorously construct an exact
solution of the quantum $SL(2)$ WZW model on lattice.
