Derivations on the triplet $W$-algebras with $\mathfrak{sl}_2$-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 Adamovi\'{c}-Milas derivation for $\mathcal{W}_{2,p}$ extends to our derivations. As an application, we show that the $\mathfrak{sl}_2$-symmetry of $\mathcal{W}_{p_+,p_-}$ arises naturally from our construction. We further show that our method applies to the triplet $W$-superalgebra $\mathcal{SW}(m)$ and that the full automorphism group ${\rm Aut}(\mathcal{SW}(m))$ is $PSL_2(\mathbb{C})\times \mathbb{Z}_2$.