Braid group symmetries on quasi-split $\mathrm{ı}$quantum groups via $\mathrm{ı}$Hall algebras

We establish automorphisms with closed formulas on quasi-split $\mathrm{ı}$quantum groups of symmetric Kac-Moody type associated to restricted Weyl groups. The proofs are carried out in the framework of $\mathrm{ı}$Hall algebras and reflection functors, thanks to the $\mathrm{ı}$Hall algebra realization of $\mathrm{ı}$quantum groups in our previous work. Several quantum binomial identities arising along the way are established.