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

TL;DR

This paper constructs a braid group action and root vectors for a specific affine $ extit{i}$-quantum group, establishing a Drinfeld type presentation for the real rank one case, advancing the understanding of affine quantum symmetric pairs.

Contribution

It explicitly constructs a braid group action, root vectors, and a Drinfeld type presentation for the affine $ extit{i}$-quantum group in the real rank one case, providing foundational tools for higher rank cases.

Findings

Constructed a relative braid group action of type A2^{(2)}.

Built real and imaginary root vectors for the affine $ extit{i}$-quantum group.

Established a Drinfeld type presentation for the affine $ extit{i}$-quantum group.

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.