For the split symmetric pair so_m ⊂ sl_m, the monodromy of the boundary Casimir connection is isomorphic to the iota-quantum Weyl group representation.
The crystal commutor and Drinfeld's unitarized R-matrix
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Drinfeld defined a unitarized R-matrix for any quantum group U_q(g). This gives a commutor for the category of U_q(g) representations, making it into a coboundary category. Henriques and Kamnitzer defined another commutor which also gives U_q(g) representations the structure of a coboundary category. We show that a particular case of Henriques and Kamnitzer's construction agrees with Drinfeld's commutor. We then describe the action of Drinfeld's commutor on a tensor product of two crystal bases, and explain the relation to the crystal commutor.
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A Kohno--Drinfeld Theorem for iquantum Weyl groups
For the split symmetric pair so_m ⊂ sl_m, the monodromy of the boundary Casimir connection is isomorphic to the iota-quantum Weyl group representation.