A Kazhdan–Lusztig Correspondence for L-32(sl3)

Abstract

The abelian and monoidal structure of the category of smooth weight modules over a non-integrable affine vertex algebra of rank greater than one is an interesting, difficult and essentially wide open problem. Even conjectures are lacking. This work details and tests such a conjecture for L-32(sl3) via a logarithmic Kazhdan–Lusztig correspondence. We first investigate the representation theory of U¯iH(sl3), the unrolled restricted quantum group of sl3 at fourth root of unity. In particular, we analyse its finite-dimensional weight category, determining Loewy diagrams for all projective indecomposables and decomposing all tensor products of irreducibles. Our motivation is that this category is..