  Poisson-Lie (PL) dynamical r-matrices are generalizations of dynamical
r-matrices, where the base is a Poisson-Lie group. We prove analogues of basic
results for these r-matrices, namely constructions of (quasi)Poisson groupoids
and of Poisson homogeneous spaces. We introduce a class of PL dynamical
r-matrices, associated to nondegenerate Lie bialgebras with a splitting; this
is a generalization of trigonometric r-matrices with an abelian base. We prove
a composition theorem for PL dynamical r-matrices, and construct quantizations
of the polarized PL dynamical r-matrices. This way, we obtain quantizations of
Poisson homogeneous structures on G/L (G a semisimple Lie group, L a Levi
subgroup), thereby generalizing earlier constructions.
