Kazhdan-Lusztig-dual quantum group for logarithimic extensions of Virasoro minimal models

We derive and study a quantum group g(p,q) that is Kazhdan-Lusztig dual to the W-algebra W-p,W-q of the logarithmic (p,q) conformal field theory model. The algebra W-p,W-q is generated by two currents W+(z) and W-(z) of dimension (2p-1)(2q-1) and the energy-momentum tensor T(z). The two currents generate a vertex-operator ideal R with the property that the quotient W-p,W-q/R is the vertex-operator algebra of the (p,q) Virasoro minimal model. The number (2pq) of irreducible g(p,q) representations is the same as the number of irreducible W-p,W-q representations on which R acts nontrivially. We find the center of g(p,q) and show that the modular group representation on it is equivalent to the modular group representation on the W-p,W-q characters and "pseudocharacters." The factorization of the g(p,q) ribbon element leads to a factorization of the modular group representation on the center. We also find the g(p,q) Grothendieck ring, which is presumably the "logarithmic" fusion of the (p,q) model.