The classification of Poisson homogeneous spaces of compact Poisson–Lie group

The classification of all Poisson homogeneous spaces with connected stabilizers of compact Poisson–Lie group $K$ (equipped with the standard $r$-matrix Poisson structure) is given. The connected closed subgroups $H\subset K$ such that $K/H$ admits a structure of Poisson homogeneous $K$-space are listed. The geometric interpretation of some of Poisson homogeneous $K$-spaces is also given.