For every finite type quantum symmetric pair, the braid group action and a PBW basis descend to the DCKP-type integral form and, via semiclassical limits, yield Poisson automorphisms and explicit Poisson brackets on the Poisson homogeneous space.
PBW bases for $\imath$quantum groups
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abstract
We establish PBW type bases for $\imath$quantum groups of arbitrary finite type, using the relative braid group symmetries. Explicit formulas for root vectors are provided for $\imath$quantum groups of each rank 1 type. We show that our PBW type bases give rise to integral bases for the modified $\imath$quantum groups. The leading terms of our bases can be identified with the usual PBW bases in the theory of quantum groups.
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Braid group symmetries on Poisson algebras arising from quantum symmetric pairs
For every finite type quantum symmetric pair, the braid group action and a PBW basis descend to the DCKP-type integral form and, via semiclassical limits, yield Poisson automorphisms and explicit Poisson brackets on the Poisson homogeneous space.