Precise Kazhdan–Lusztig duality for W_{p,q}

Establish a precise version of the conjectured relation between the monoidal category of modules for the triplet vertex operator algebra W_{p,q} (for coprime integers p,q ≥ 2) and the representation category of the corresponding Kazhdan–Lusztig dual quantum group, generalizing the proven correspondence in the case q = 1.

Background

For q = 1, the representation category of the triplet vertex operator algebra W_{p,1} is known to be equivalent to the category of finite-dimensional representations of a quasi-Hopf modification of the restricted quantum group of sl2 at a root of unity. For general coprime p,q ≥ 2, Feigin–Gainutdinov–Semikhatov–Tipunin conjectured a related duality with a suitable “Kazhdan–Lusztig dual” quantum group, but only a relation (not a full equivalence) was proposed.

The paper highlights that turning this conjectural relation into a precise equivalence (or otherwise fully characterizing it) remains a significant unresolved issue for the W_{p,q} triplet algebra, and resolving it would extend the established q=1 paradigm to the broader logarithmic W-algebra context.