Braid group action and quasi-split affine 𝚤quantum groups I

Abstract

This is the first of our papers on quasi-split affine quantum symmetric pairs ( U ~ ( g ^ ) , U ~ ı ) \big (\widetilde {\mathbf U}(\widehat {\mathfrak g}), \widetilde {{\mathbf U}}^\imath \big ) , focusing on the real rank one case, i.e., g = s l 3 \mathfrak g = \mathfrak {sl}_3 equipped with a diagram involution. We construct explicitly a relative braid group action of type A 2 ( 2 ) A_2^{(2)} on the affine ı \imath quantum group U ~ ı \widetilde {{\mathbf U}}^\imath . Real and imaginary root vectors for U ~ ı \widetilde {{\mathbf U}}^\imath are constructed, and a Drinfeld type presentation of U ~ ı \widetilde {{\mathbf U}}^\imath is then established. This provides a new basic ingredient for the Drinfeld type presentation of higher rank quasi-split affine ı \imath quantum groups in the sequels.