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

TL;DR

This work establishes a Drinfeld-type presentation for the quasi-split affine $\imath$quantum group $\widetilde{{\mathbf U}}^{\imath}$ in the real rank one setting $\mathfrak{g}=\mathfrak{sl}_3$ with a diagram involution. It constructs a relative braid group action of twisted type $A_2^{(2)}$, builds real and imaginary root vectors, and proves a current-Drinfeld presentation via generating-function relations, with a rigorous isomorphism between the Drinfeld model ${}^{\text{Dr}}\widetilde{{\mathbf U}}^{\imath}$ and the original algebra $\widetilde{{\mathbf U}}^{\imath}$. The approach combines an intertwining property with a rank-one quasi-$K$-matrix, a translation-generated root-vector framework, and a detailed verification of all Drinfeld-type relations, providing a foundational toolkit for higher-rank quasi-split affine $\imath$quantum groups. The results are expected to underpin representation theory and integrable systems in the quasi-split affine setting and to guide future generalizations.

Abstract

This is the first of two 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 sequel.

Theorems & Definitions (101)

• Proposition 2.1
• Lemma 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• Proposition 2.5: WZ22
• proof
• Lemma 2.6: DK19
• Theorem 2.7
• Theorem 2.8