Braid group action and quasi-split affine $\imath$quantum groups I

Abstract:

This is the first of our papers on quasi-split affine quantum symmetric pairs $\big (\widetilde {\mathbf U}(\widehat {\mathfrak g}), \widetilde {{\mathbf U}}^\imath \big )$, focusing on the real rank one case, i.e., $\mathfrak g = \mathfrak {sl}_3$ equipped with a diagram involution. We construct explicitly a relative braid group action of type $A_2^{(2)}$ on the affine $\imath$quantum group $\widetilde {{\mathbf U}}^\imath$. Real and imaginary root vectors for $\widetilde {{\mathbf U}}^\imath$ are constructed, and a Drinfeld type presentation of $\widetilde {{\mathbf U}}^\imath$ is then established. This provides a new basic ingredient for the Drinfeld type presentation of higher rank quasi-split affine $\imath$quantum groups in the sequels.