Affinized braid group action in full generality

Establish an action of the affinized braid group \widehat{B} on the quantum affinization \widehat{U_q(\mathfrak{s})} for arbitrary symmetrizable Kac–Moody algebras \mathfrak{s}, removing the assumption of condition (D) on the Dynkin diagram.

Background

Under condition (D), the paper constructs an affinized braid group action on quantum affinizations, extending Lusztig–Beck actions and enabling key symmetry constructions used later (e.g., horizontal–vertical symmetries).

A general action without condition (D) would strengthen the framework, allowing the same symmetry-based techniques for all quantum affinizations.