Abstract
Global issues of the Poisson-Lie T-duality are addressed. It is shown that oriented open strings propagating on a group manifold
G
are dual to
D
-brane - anti-
D
-brane pairs propagating on the dual group manifold $\ti G$. The
D
-branes coincide with the symplectic leaves of the standard Poisson structure induced on the dual group $\ti G$ by the dressing action of the group
G
. T-duality maps the momentum of the open string into the mutual distance of the
D
-branes in the pair. The whole picture is then extended to the full modular space
M(D)
of the Poisson-Lie equivalent $\si$-models which is the space of all Manin triples of a given Drinfeld double.T-duality rotates the zero modes of pairs of
D
-branes living on targets belonging to
M(D)
. In this more general case the
D
-branes are preimages of symplectic leaves in certain Poisson homogeneous spaces of their targets and, as such, they are either all even or all odd dimensional.