Joseph ideals and lisse minimal W-algebras

TL;DR

The paper advances the understanding of the geometric and representation-theoretic structure of affine vertex algebras by lifting the Joseph ideal associated with the minimal nilpotent orbit to the affine setting, yielding new examples where the associated variety is the minimal nilpotent orbit closure. It develops a framework in which $V_k(rak g)$ serves as a chiralization of Joseph-type quotients, and constructs explicit degree-2 singular vectors to analyze simple quotients and their associated varieties. Using BRST reduction, the authors produce lisse $W$-algebras $rak W_k(rak g,f_ heta)$ not arising from admissible representations, clarifying when lisse minimal $W$-algebras occur across Deligne exceptional and classical types. They culminate with a classification of lisse minimal $W$-algebras, detailing level- and type-dependent conditions and highlighting new non-admissible examples with potential rationality implications. The results provide both new counterpoints to conjectures linking admissibility to lisseness and a comprehensive landscape of minimal $W$-algebras in the affine setting.

Abstract

We consider a lifting of Joseph ideals for the minimal nilpotent orbit closure to the setting of affine Kac-Moody algebras and find new examples of affine vertex algebras whose associated varieties are minimal nilpotent orbit closures. As an application we obtain a new family of lisse ($C_2$-cofinite) W-algebras that are not coming from admissible representations of affine Kac-Moody algebras.

Theorems & Definitions (37)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Proposition 2.2
• proof
• Theorem 3.1
• Lemma 4.1
• Theorem 4.2
• proof