Rank-one formulas for relative braid group symmetries preserving integral forms

Construct explicit rank-one formulas for the relative braid group symmetries acting on all integrable modules over a quasi-split iquantum group _\bvs and on the modified iquantum group \dot{U}^\imath, such that these formulas preserve the integral \mathbb{Z}[q,q^{-1}] forms of the modules and of \dot{U}^\imath.

Background

Lusztig’s rank-one braid group formulas for quantum groups are fundamental and directly applicable to modules, but analogous explicit formulas for iquantum groups were not available. Earlier constructions of relative braid group symmetries for iquantum groups relied on quasi K-matrices or Hall algebra realizations and either did not yield module actions or were not explicit enough for computations.

This problem seeks formulas that work for all integrable modules over quasi-split iquantum groups and for the modified iquantum group itself, with the additional requirement that they respect integral \mathbb{Z}[q,q^{-1}] forms—an essential property for canonical and icanonical bases and for arithmetic applications.