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Joseph ideals and lisse minimal \(w\)-algebras. (English) Zbl 1416.17014

Summary: 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.

MSC:

17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
17B69 Vertex operators; vertex operator algebras and related structures
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
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