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

Abstract

This is the first of two papers on quasi-split affine quantum symmetric pairs $(\tilde{\mathbf{U}}(\hat{\mathfrak{g}}),{\tilde{\mathbf{U}}}^{ı})$, 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 $ı$quantum group ${\tilde{\mathbf{U}}}^{ı}$. Real and imaginary root vectors for ${\tilde{\mathbf{U}}}^{ı}$ are constructed, and a Drinfeld type presentation of ${\tilde{\mathbf{U}}}^{ı}$ is then established. This provides a new basic ingredient for the Drinfeld type presentation of higher rank quasi-split affine $ı$quantum groups in the sequel.

v3, added factors 1/(v-v^{-1}) to the RHS of (5.27) missed in the published version, 43 pages