  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.
