  In order to study quantum aspects of $\s$-models related by Poisson--Lie
T-duality, we construct three- and two-dimensional models that correspond, in
one of the dual faces, to deformations of $S^3$ and $S^2$. Their classical
canonical equivalence is demonstrated by means of a generating functional,
which we explicitly compute. We examine how they behave under the
renormalization group and show that dually related models have the same 1-loop
beta functions for the coupling and deformation parameters. We find non-trivial
fixed points in the ultraviolet, where the theories do not become
asymptotically free. This suggests that the limit of Poisson--Lie T-duality to
the usual Abelian and non-Abelian T-dualities does not exist quantum
mechanically, although it does so classically.
