A realisation of the Bershadsky--Polyakov algebras and their relaxed modules

Abstract

We present a realisation of the universal/simple Bershadsky--Polyakov vertex algebras as subalgebras of the tensor product of the universal/simple Zamolodchikov vertex algebras and an isotropic lattice vertex algebra. This generalises the realisation of the universal/simple affine vertex algebras associated to ${\mathfrak{sl}}_2$ and $\mathfrak{osp}(1|2)$ given in arXiv:1711.11342. Relaxed highest-weight modules are likewise constructed, conditions for their irreducibility are established, and their characters are explicitly computed, generalising the character formulae of arXiv:1803.01989.

arXiv comment23 pages; v2 24 pages --- added a few more details and remarks, fixed a few typos; to appear in Lett. Math. Phys

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Cited by in corpus (13)

• Admissible-level ${\mathfrak{sl}}_3$ minimal models
• Modularity of Bershadsky-Polyakov minimal models
• Subregular W-algebras of type A
• Rigid tensor structure on big module categories for some $W$-(super)algebras in type $A$
• On the semisimplicity of the category $K{L}_k$ for affine Lie superalgebras
• Representations of the Nappi--Witten vertex operator algebra
• Reduction by stages for finite W-algebras
• On the structure of W-algebras in type A
• Orthosymplectic Feigin-Semikhatov duality
• Feigin-Semikhatov conjecture and related topics
• Associated varieties of simple affine VOAs $L_k(s{l}_3)$ and $W$-algebras $W_k(s{l}_3,f)$
• Connecting affine $\mathcal{W}$-algebras: A case study on ${\mathfrak{sl}}_4$
• Relaxed and logarithmic modules of $\widehat{{\mathfrak{sl}}_3}$