  We provide a ribbon tensor equivalence between the representation category of
small quantum SL(2), at parameter q=exp($\pi$ i/p), and the representation
category of the triplet vertex operator algebra at integral parameter p>1. We
provide similar quantum group equivalences for representation categories
associated to the Virasoro, and singlet vertex operator algebras at central
charge c=1-6(p-1)^2/p. These results resolve a number of fundamental
conjectures coming from studies of logarithmic CFTs in type A_1.
