Abstract
We study the simple Bershadsky–Polyakov algebra 𝒲k = 𝒲k(sl3, fθ) at positive integer levels and classify their irreducible modules. In this way, we confirm the conjecture from [9]. Next, we study the case k = 1. We discover that this vertex algebra has a Kazama–Suzuki-type dual isomorphic to the simple affine vertex superalgebra Lk′ (osp(1|2)) for k′ = –5=4. Using the free-field realization of Lk′ (osp(1|2)) from [3], we get a free-field realization of 𝒲k and their highest weight modules. In a sequel, we plan to study fusion rules for 𝒲k.
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