  The quantum duality principal (QDP) by Drinfeld predicts a connection between
the quantized universial enveloping algebras and the quantized coordinate
algebras, where the underlying classical objects are related by the duality in
Poisson geometry. The current paper gives an explicit formulization of the QDP
for quantum symmetric pairs.
  Let $\mathfrak{g}$ be a complex semi-simple Lie algebra, equipped with the
standard Lie bialgebra structure. Let $\theta$ be a Lie algebra involution on
$\mathfrak{g}$ and denote by $\mathfrak{k}=\mathfrak{g}^\theta$ the fixed point
subalgebra. The quantum symmetric pair $(\mathrm{U},\mathrm{U}^\imath)$ is
originally defined to be a quantization of the symmetric pair of the universial
enveloping algebras $(U(\mathfrak{g}),U(\mathfrak{k}))$. In this paper, we show
that an explicit specialisation of $(\mathrm{U},\mathrm{U}^\imath)$ gives rise
to the pair of the coordinate algebras
$(\mathcal{O}(G^*),\mathcal{O}(K^\perp\backslash G^*))$, where $G^*$ is the
dual Poisson-Lie group with the Lie algebra $\mathfrak{g}^*$, and
$K^\perp\backslash G^*$ is a $G^*$-Poisson homogeneous space. Here $K^\perp$ is
the closed subgroup of $G^*$associated to the complementary dual of
$\mathfrak{k}$. Therefore $(\mathrm{U},\mathrm{U}^\imath)$ can be viewed as a
pair of quantized coordinate algebras. This generalises the well-known fact
that the quantum group $\mathrm{U}$ provides a quantization of the coordinate
algebra $\mathcal{O}(G^*)$.
