Relative braid group symmetries on modified iquantum groups and their modules

TL;DR

This paper generalizes Lusztig's braid group symmetries to quasi-split iquantum groups, providing explicit formulas and demonstrating their consistency and relations with existing symmetries across various modules.

Contribution

It introduces explicit rank one formulas for symmetries on integrable modules over quasi-split iquantum groups of arbitrary Kac-Moody type, expanding the understanding of their symmetry structures.

Findings

Symmetries are consistent with relative braid group symmetries.

Explicit formulas are provided for rank one cases.

Symmetries satisfy the relative braid group relations.

Abstract

We present a comprehensive generalization of Lusztig's braid group symmetries for quasi-split iquantum groups. Specifically, we give 3 explicit rank one formulas for symmetries acting on integrable modules over a quasi-split iquantum group of arbitrary Kac-Moody type with general parameters. These symmetries are formulated in terms of idivided powers and iweights of the vectors being acted upon. We show that these symmetries are consistent with the relative braid group symmetries on iquantum groups, and they are also related to Lusztig's symmetries via quasi $K$-matrices. Furthermore, through appropriate rescaling, we obtain compatible symmetries for the integral forms of modified iquantum groups and their integrable modules. We verify that these symmetries satisfy the relative braid group relations.