The symplectic leaves on Poisson homogeneous spaces of Poisson–Lie groups

The symplectic leaves on Poisson homogeneous spaces of a Poisson–Lie group $G$ are described in terms of the double $D(G)$. An explicit formula for the symplectic form on a leaf is derived. A duality between the symplectic leaves on the Poisson homogeneous spaces of $G$ and those of the dual group $G^*$ is revealed.