$N=2$ supersymmetric structures on classical $W$-algebras

We describe an $N=2$ supersymmetric Poisson vertex algebra structure of $N=1$ (resp. $N=0$) classical $W$-algebra associated with $\mathfrak{s}\mathfrak{l}(n+1\mathrm{\mid }n)$ and the odd (resp. even) principal nilpotent element. This $N=2$ supersymmetric structure is connected to the principal $\mathfrak{s}\mathfrak{l}(2\mathrm{\mid }1)$-embedding in $\mathfrak{s}\mathfrak{l}(n+1\mathrm{\mid }n)$ superalgebras, which are the only basic Lie superalgebras that admit such a principal embedding.