  We introduce a Kazhdan--Lusztig-dual quantum group for (1,p) Virasoro
logarithmic minimal models as the Lusztig limit of the quantum sl(2) at pth
root of unity and show that this limit is a Hopf algebra. We calculate tensor
products of irreducible and projective representations of the quantum group and
show that these tensor products coincide with the fusion of irreducible and
logarithmic modules in the (1,p) Virasoro logarithmic minimal models.
