Weight module classifications for Bershadsky--Polyakov algebras

The Bershadsky–Polyakov algebras are the subregular quantum hamiltonian reductions of the affine vertex operator algebras associated with ${\mathfrak{s}\mathfrak{l}}_3$. In arXiv:2007.00396 [math.QA], we realised these algebras in terms of the regular reduction, Zamolodchikov’s W${}_3$-algebra, and an isotropic lattice vertex operator algebra. We also proved that a natural construction of relaxed highest-weight Bershadsky–Polyakov modules gives modules that are generically irreducible. Here, we prove that this construction, when combined with spectral flow twists, gives a complete set of irreducible weight modules whose weight spaces are finite-dimensional. This gives a simple independent proof of the main classification theorem of arXiv:2007.03917 [math.RT] for nondegenerate admissible levels and extends this classification to a category of weight modules. We also deduce the classification for the nonadmissible level $\mathsf{k}=-\frac{7}{3}$, which is new.