Braid group symmetries on Poisson algebras arising from quantum symmetric pairs

Let $(\mathrm{U},\mathrm{U}^\imath)$ be the quantum symmetric pair of arbitrary finite type and $G^*$ be the associated dual Poisson-Lie group. Generalizing the work of De Concini and Procesi, the first author introduced an integral form for the $\imath$quantum group $\mathrm{U}^\imath$ and its semi-classical limit was shown to be the coordinate algebra for a Poisson homogeneous space of $G^*$. In this paper, we establish (relative) braid group symmetries and PBW bases on this integral form of $\mathrm{U}^\imath$. By taking the semi-classical limit, we obtain braid group symmetries and polynomial generators on the associated Poisson algebra. These symmetries further allow us to describe the Poisson brackets explicitly. Examples of such Poisson structures include Dubrovin-Ugaglia Poisson brackets.