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

Abstract

To study the representation category of the triplet W-algebra \(\mathcal{W}\left( p \right)\) that is the symmetry of the (1, p) logarithmic conformal field theory model, we propose the equivalent category Cp of finite-dimensional representations of the restricted quantum group Ūqsl(2) at \(\mathfrak{q} = e^{{{i\pi } \mathord{\left/ {\vphantom {{i\pi } p}} \right. \kern-\nulldelimiterspace} p}} \). We fully describe the category Cp 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 \(\mathcal{W}\left( p \right)\)-and Ūqsl(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.