The triplet vertex operator algebra $W(p)$ and the restricted quantum group $\bar{U}_q (sl_2)$ at $q = e^{\frac{\pi i}{p}}$

Abstract

We study the abelian category $W(p)$-mod of modules over the triplet $W$ algebra $W(p)$. We construct the projective covers $\mathcal{P}_s^{\pm}$ of all the simple objects $\mathcal{X}_s^{\pm}$, $1 \leq s \leq p$, in the category $W(p)$-mod. By using the structure of these projective modules, we show that $W(p)$-mod is a category which is equivalent to the abelian category of the finite-dimensional modules for the restricted quantum group $\bar{U}_q (sl_2)$ at $q = e^{\frac{\pi i}{p}}$. This Kazdan–Lusztig type correspondence was conjectured by Feigin et al. [FGST1], [FGST2].