Finite presentation of quantum affinizations beyond condition (D)

Determine whether every quantum affinization \widehat{U_q(\mathfrak{s})} associated to a symmetrizable Kac–Moody algebra \mathfrak{s} admits a finite presentation (by a finite set of generators and relations) without imposing condition (D) on the Cartan matrix (i.e., allowing arbitrary multiple arrows).

Background

The paper proves a finite presentation for quantum affinizations under condition (D), which requires that all pairwise products a_{ij}a_{ji} are at most 3 or both equal to −2. This enables the construction of braid group actions and various structural results.

Extending a finite presentation to all quantum affinizations (without condition (D)) would significantly broaden applicability, including to types with higher multiplicity of arrows in the Dynkin diagram.