Simplicity of Higher Rank Triplet W-Algebras

Abstract

Abstract We show that the higher rank triplet $W$-algebra $W_{\sqrt {p}Q}$ is simple for $p\geq h-1$. Furthermore, we show that the $W_{\sqrt {p}Q}$-module $W_{\sqrt {p}(Q-\lambda _0)+\lambda _p}$ introduced in [21] is simple if $\sqrt {p}\lambda _p$ is in the closure of the fundamental alcove and give the decomposition as a direct sum of simple $\boldsymbol {W}_{p-h}({\mathfrak {g}})$-modules.