Explicit constructions of irreducible quantum-group modules at roots of unity

Construct explicit irreducible modules for quantum groups U_q(𝔤) at roots of unity in full generality.

Background

The paper develops Gelfand–Tsetlin bases for representations of U_q(gl_n) when q is a root of unity and applies them to selected baby Verma modules. Despite these constructions, the authors identify a broader unresolved issue: explicit constructions of irreducible modules for quantum groups at roots of unity are not available in full generality. The problem therefore concerns extending explicit representation-theoretic constructions beyond the families treated in the paper.