Are the odd dimensional spheres Poisson homogeneous spaces? Are the odd dimensional spheres $S^{2n+1}$, for $n \in \mathbb{N}_{\geq 1}$, Poisson homogeneous spaces in the sense of Drinfeld?
 A: The question is not meaningful unless you clarify which Poisson structure on $\mathbb S^{n+1}$ you are considering; any homogeneous space is Poisson-homogeneous when endowed with the null Poisson bracket with respect to the null Poisson-Lie group structure.
But the answer is yes in a more reasonable sense; say you consider the standard compact Poisson-Lie group structure on $SU(n+1)$. Then $SU(n)$ embedded either as upper left corner or lower right corner is a Poisson-Lie subgroup. Therefore the quotient $\mathbb S^{2n+1}\simeq SU(n+1)/SU(N)$ is naturally endowed with a Poisson structure making it into a Poisson homogeneous space.
The corresponding symplectic foliation is well understood. There are $\mathbb S^1$-families of simply connected symplectic leaves in each even dimension, each bounded by lower-dimensional leaves (so that in fact this singular foliation resembles more a stratification).
I do not know whether this is the unique realization of odd-spheres as (non trivial) Poisson homogeneous $G$-space; I guess it is not.

Of course it is not. Fo example taking a so-called twisted Poisson-Lie group structure on $SU(n)$ one has that the usual embedding of $SO(n)$ is coisotropic and therefore any sphere $SU(n)/SO(n)$ has a Poisson homogeneous structure (this time w. respect to a different Poisson-Lie group structure on $SU(n)$ , generic leaves are now tori).
