Home > Proceedings > Adv. Stud. Pure Math. > Exploring New Structures and Natural Constructions in Mathematical Physics > The triplet vertex operator algebra $W(p)$ and the restricted quantum group $\bar{U}_q (sl_2)$ at $q = e^{\frac{\pi i}{p}}$
Translator Disclaimer
VOL. 61 | 2011 The triplet vertex operator algebra $W(p)$ and the restricted quantum group $\bar{U}_q (sl_2)$ at $q = e^{\frac{\pi i}{p}}$
Kiyokazu Nagatomo, Akihiro Tsuchiya

Editor(s) Koji Hasegawa, Takahiro Hayashi, Shinobu Hosono, Yasuhiko Yamada

## 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].

## Information

Published: 1 January 2011
First available in Project Euclid: 24 November 2018

zbMATH: 1247.81217
MathSciNet: MR2867143

Digital Object Identifier: 10.2969/aspm/06110001