  This work is motivated by a result of Drinfeld on Poisson homogeneous spaces.
For each Poisson manifold $P$ with a Poisson action by a Poisson Lie group $G$,
we describe a Lie algebroid structure on the direct sum vector bundle $P \times
{\frak g} \oplus T^*P$, where ${\frak g}$ is the Lie algebra of $G$. It is
built out of the transformation Lie algebroid $P \times {\frak g}$ and the
cotangent bundle Lie algebroid $T^*P$ together with a pair of representations
of them on each other. When the action of $G$ on $P$ is transitive, the kernel
of the anchor map of this Lie algebroid gives a Lie algebra bundle over $P$,
the fibers of which are given by Drinfeld. As applications, we describe the
symplectic leaves and the $G$-invariant Poisson cohomology of Poisson
homogeneous $G$-spaces.
