Simplicity of higher rank triplet W-algebras

TL;DR

This paper proves the simplicity of higher rank triplet W-algebras for certain parameters and analyzes the simplicity and decomposition of associated modules, advancing understanding of their algebraic structure.

Contribution

It establishes the simplicity conditions for higher rank triplet W-algebras and their modules, providing new insights into their representation theory.

Findings

Higher rank triplet W-algebra is simple for p ≥ h-1

Feigin-Tipunin modules are simple within the closure of the fundamental alcove

Modules decompose into direct sums of simple modules over affine W-algebras

Abstract

We show that the higher rank triplet W-algebra is simple when p is bigger than or equal to h-1. Furthermore, we show that the Feigin-Tipunin's module over the higher rank triplet W-algebra introduced in arXiv:1002.5047 is simple if the parameter is in the closure of the fundamental alcove, and give the decomposition as a direct sum of simple modules over the affine W-algebras at level p-h.