  We propose a Poisson-Lie analog of the symplectic induction procedure, using
an appropriate Poisson generalization of the reduction of symplectic manifolds
with symmetry. Having as basic tools the equivariant momentum maps of Poisson
actions, the double group of a Poisson-Lie group and the reduction of Poisson
manifolds with symmetry, we show how one can induce a Poisson action admitting
an equivariant momentum map. We prove that, under certain conditions, the
dressing orbits of a Poisson-Lie group can be obtained by Poisson induction
from the dressing orbits of a Poisson-Lie subgroup.
