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

TL;DR

This paper constructs realizations of Bershadsky--Polyakov vertex algebras using tensor products of Zamolodchikov algebras and lattice vertex algebras, extending previous affine algebra realizations, and develops their relaxed modules with explicit character formulas.

Contribution

It generalizes the realization of Bershadsky--Polyakov algebras and constructs their relaxed modules with explicit character formulas.

Findings

Realization of Bershadsky--Polyakov algebras as subalgebras of tensor products.

Construction of relaxed highest-weight modules.

Explicit character formulas for these 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.