Derivations on the triplet $W$-algebras with ${\mathfrak{s}\mathfrak{l}}_2$-symmetry

We construct derivations on the triplet $W$-algebras ${\mathcal{W}}_{p_{+},p_{-}}$ by refining the Frobenius homomorphisms of Tsuchiya-Wood and show that the property of the Adamović-Milas derivation for ${\mathcal{W}}_{2,p}$ extends to our derivations. As an application, we show that the ${\mathfrak{s}\mathfrak{l}}_2$-symmetry of ${\mathcal{W}}_{p_{+},p_{-}}$ arises naturally from our construction. We further show that our method applies to the triplet $W$-superalgebra $\mathcal{S}\mathcal{W}(m)$ and that the full automorphism group $\mathrm{A}\mathrm{u}\mathrm{t}(\mathcal{S}\mathcal{W}(m))$ is $PS{L}_2(\mathbb{C})\times {\mathbb{Z}}_2$.