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 $\boldsymbol{\mathcal{W}}(p)$ that is the symmetry of the $(1,p)$ logarithmic conformal field theory model, we propose the equivalent category $\EuScript{C}_p$ of finite-dimensional representations of the restricted quantum group $\overline{\EuScript{U}}_{\!\mathfrak{q}} s\ell(2)$ at $\mathfrak{q}=e^{{i\pi}/{p}}$. We fully describe the category $\EuScript{C}_p$ 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 $\boldsymbol{\mathcal{W}}(p)$- and $\overline{\EuScript{U}}_{\!\mathfrak{q}} s\ell(2)$-representation categories is conjectured for all $p\ge2$ 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.