General classification of finite-dimensional simple modules for finite-type i-quantum groups

Develop a general classification theory for the finite-dimensional simple modules over quantum symmetric pair coideal subalgebras U^i of finite type (i-quantum groups), providing a unified framework that classifies all such U^i-modules across finite types.

Background

The paper highlights that, despite substantial progress on quantum groups, the representation theory of i-quantum groups (quantum symmetric pair coideal subalgebras) has advanced slowly. Unlike the well-developed classification for finite-dimensional simple modules over quantum groups, no general theory exists yet for i-quantum groups of finite type.

Existing results are type-dependent and scattered across specific cases, as referenced in the literature (e.g., works by Gavrilik–Klimyk, Iorgov–Klimyk, Molev, Ito–Terwilliger, Watanabe, Wenzl, Kolb–Stephens). This paper introduces integrable modules for i-quantum groups and develops structural tools, but the comprehensive classification problem remains unresolved.