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We study the structure of the abelian category of modules for the triplet$W$-algebra $\mathcal{W}_{p_+,p_-}$. Using the logarithmic deformation byFjelstad et al.(2002), we construct logarithmic $\mathcal{W}_{p_+,p_-}$-modulesthat have $L_0$ nilpotent rank three or two. By using the structure of theselogarithmic modules and the results on logarithmic Virasoro modules byKyt\"{o}l\"{a} and Ridout(2009), we compute ${\rm Ext}^1$ groups betweencertain indecomposable modules and simple modules. Based on these ${\rm Ext}^1$groups we determine the structure of the projective covers of all$\mathcal{W}_{p_+,p_-}$-simple modules.

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