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

摘要

To study the representation category of the triplet W-algebra W(p) that is the symmetry of the (I, p) logarithmic conformal field theory model, we propose the equivalent category C, of finite-dimensional representations of the restricted quantum group (U) over bar (q)sl(2) at q = e(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) over bar (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 held theory center and the universal R-matrix with the braiding matrix.