Projective covers of the simple modules for the triplet $W$-algebra ${\mathcal{W}}_{p_{+},p_{-}}$

Abstract

We study the structure of the abelian category of modules for the triplet $W$-algebra ${\mathcal{W}}_{p_{+},p_{-}}$. Using the logarithmic deformation by Fjelstad et al., we construct logarithmic ${\mathcal{W}}_{p_{+},p_{-}}$-modules that have $L_0$ nilpotent rank three or two. By using the structure of these logarithmic modules and the results on logarithmic Virasoro modules by Kytölä and Ridout, we compute ${\mathrm{Ext}}^1$ groups between certain indecomposable modules and simple modules. Based on these ${\mathrm{Ext}}^1$ groups we determine the structure of the projective covers of all ${\mathcal{W}}_{p_{+},p_{-}}$-simple modules.

arXiv comment86 pages, 13 figures. Revised the figures, improved the notation and several proofs, and included a proof concerning the block decomposition