Kazhdan-Lusztig correspondence for the representation category of the triplet W-algebra in logarithmic CFT

To study the representation category of the triplet W-algebra W ( p)that is the symmetry of the (1, p) logarithmic conformal field theory model, we propose the equivalent category C_p of finite-dimensional representations of the restricted quantum group Ū _qsl(2) at q = e iφ/p. We fully describe the category C _p by classifying all indecomposable representations. These are exhausted by projective modules and three series of representations that are essentially described by indecomposable representations of the Kronecker quiver. The equivalence of the W (p)- and Ū _q sl(2)-representation categories is conjectured for all p = 2 and proved for p = 2. The implications include identifying the quantum group center with the logarithmic conformal field theory center and the universal R-matrix with the braiding matrix.