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

We derive and study a quantum group $gp,q$ that is Kazhdan-Lusztig dual to the $W$-algebra $Wp,q$ of the logarithmic $(p,q)$ conformal field theory model. The algebra $Wp,q$ 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 $Wp,q∕R$ is the vertex-operator algebra of the $(p,q)$ Virasoro minimal model. The number $(2pq)$ of irreducible $gp,q$ representations is the same as the number of irreducible $Wp,q$ representations on which $R$ acts nontrivially. We find the center of $gp,q$ and show that the modular group representation on it is equivalent to the modular group representation on the $Wp,q$ characters and “pseudocharacters.” The factorization of the $gp,q$ ribbon element leads to a factorization of the modular group representation on the center. We also find the $gp,q$ Grothendieck ring, which is presumably the “logarithmic” fusion of the $(p,q)$ model.