  We identify the cotangent bundle Lie algebroid of a Poisson homogeneous space
G/H of a Poisson Lie group G as a quotient of a transformation Lie algebroid
over G. As applications, we describe the modular vector fields of G/H, and we
identify the Poisson cohomology of G/H with coefficients in powers of its
canonical line bundle with relative Lie algebra cohomology of the Drinfeld Lie
algebra associated to G/H. We also construct a Poisson groupoid over G/H which
is symplectic near the identity section. This note serves as preparation for
forthcoming papers, in which we will compute explicitly the Poisson cohomology
and study their symplectic groupoids for certain examples of Poisson
homogeneous spaces related to semi-simple Lie groups.
