Kazhdan–Lusztig equivalence and fusion of Kac modules in Virasoro logarithmic models

Abstract

The subject of our study is the Kazhdan–Lusztig (KL) equivalence in the context of a one-parameter family of logarithmic CFTs based on Virasoro symmetry with the (1,p) central charge. All finite-dimensional indecomposable modules of the KL-dual quantum group — the “full” Lusztig quantum sℓ(2) at the root of unity — are explicitly described. These are exhausted by projective modules and four series of modules that have a functorial correspondence with any finitely-generated quotient or a submodule of Feigin–Fuchs modules over the Virasoro algebra. Our main result includes calculation of tensor products of any pair of the indecomposable modules. Based on the Kazhdan–Lusztig equivalence between quantum groups and vertex-operator algebras, fusion rules of Kac modules over the Virasoro algebra in the (1,p) LCFT models are conjectured.