Relative braid group symmetries on $\imath$quantum groups of Kac-Moody type

TL;DR

This paper extends the construction of relative braid group symmetries to 0imathquantum groups of Kac-Moody type, formulates root vectors, and shows their invariance under these symmetries.

Contribution

It generalizes the relative braid group actions to Kac-Moody imathquantum groups and establishes the behavior of root vectors under these symmetries.

Findings

Root vectors are preserved under relative braid group symmetries.

Formulation of root vectors in recursive and closed imathdivided power forms.

Extension of symmetry constructions to Kac-Moody types.

Abstract

Recently, relative braid group actions on $\imath$quantum groups of arbitrary finite types have been constructed by Wang and the author. In this paper, we extend that construction to $\imath$quantum groups of Kac-Moody type. We formulate root vectors for $\imath$quantum groups in both recursive forms and closed $\imath$divided power forms. We show that the relative braid group symmetries send root vectors to root vectors.