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

TL;DR

This paper constructs new derivations on triplet $W$-algebras with $rak{sl}_2$-symmetry, extending previous homomorphisms and revealing natural symmetry origins, also applying to superalgebras with a specific automorphism group.

Contribution

It refines Frobenius homomorphisms to create derivations on triplet $W$-algebras, demonstrating their $rak{sl}_2$-symmetry and extending methods to superalgebras with a known automorphism group.

Findings

Derivations on $ ext{triplet } W$-algebras constructed from refined Frobenius homomorphisms.

$rak{sl}_2$-symmetry arises naturally from the new derivations.

Method applies to $ ext{triplet } W$-superalgebra } ext{SW}(m)$ with automorphism group } PSL_2( ext{C}) imes ext{Z}_2.

Abstract

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 $PSL_2(\mathbb{C})\times \mathbb{Z}_2$.