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 $U_q SL2$ at $q=e^{\frac{i\pi}{p}}$. 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 $U_q SL2$-representation categories is conjectured for all $p\ge 2$ and proved for p=2, the implications including the identifications of the quantum-group center with the logarithmic conformal field theory center and of the universal R-matrix with the braiding matrix.