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

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.(2002), 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(2009), we compute ${\mathrm{E}\mathrm{x}\mathrm{t}}^1$ groups between certain indecomposable modules and simple modules. Based on these ${\mathrm{E}\mathrm{x}\mathrm{t}}^1$ groups we determine the structure of the projective covers of all ${\mathcal{W}}_{p_{+},p_{-}}$-simple modules.