  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 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.
