A dynamical system is canonically associated to every Drinfeld double
of any affine Kac-Moody group. The choice of the affine Lu-Weinstein-Soibelman
double gives a smooth one-parameter deformation of the standard WZW
model. In particular, the worldsheet and the target of the classical
version of the deformed theory are the ordinary smooth manifolds. The
quasitriangular WZW model is exactly solvable and it admits the chiral
decomposition. Its classical action is not invariant with respect to
the left and right action of the loop group, however it satisfies the
weaker condition of the Poisson-Lie symmetry. The structure of the deformed
WZW model is characterized by several ordinary and dynamical r-matrices
with spectral parameter. They describe the q-deformed current algebras,
they enter the definition of q-primary fields and they characterize
the quasitriangular exchange (braiding) relations. Remarkably, the symplectic
structure of the deformed chiral WZW theory is cocharacterized by the
same elliptic dynamical r-matrix that appears in the Bernard generalization
of the Knizhnik-Zamolodchikov equation, with q entering the modular
parameter of the Jacobi theta functions. This reveals a tantalizing
connection between the classical q-deformed WZW model and the quantum
standard WZW theory on elliptic curves and opens the way for the systematic
use of the dynamical Hopf algebroids in the rational q-conformal field
theory.