No abstract

The Bershadsky–Polyakov algebras are the original examples of nonregular W-algebras, obtained from the affine vertex operator algebras associated with $$\mathfrak {sl}_3$$ sl 3 by quantum Hamiltonian reduction. In Fehily et al. (Comm Math Phys 385:859–904, 2021), we explored the representation theories of the simple quotients of these algebras when the level $$\mathsf {k}$$ k is nondegenerate-admissible. Here, we combine these explorations with Adamović’s inverse quantum Hamiltonian reduction functors to study the modular properties of Bershadsky–Polyakov characters and deduce the associated Grothendieck fusion rules. The results are not dissimilar to those already known for the affine vertex operator algebras associated with $$\mathfrak {sl}_2$$ sl 2 , except that the role of the Virasoro minimal models in the latter is here played by the minimal models of Zamolodchikov’s $$\mathsf {W}_3$$ W 3 algebras.

Subregular W-algebras are an interesting and increasingly important class of quantum hamiltonian reductions of affine vertex algebras. Here, we show that the [Formula: see text] subregular W-algebra can be realized in terms of the [Formula: see text] regular W-algebra and the half lattice vertex algebra [Formula: see text]. This generalizes the realizations found for [Formula: see text] and [Formula: see text] in [D. Adamović, Realizations of simple affine vertex algebras and their modules: The cases [Formula: see text] and [Formula: see text], Comm. Math. Phys. 366 (2019) 1025–1067, arXiv:1711.11342 [math.QA]; D. Adamović, K. Kawasetsu and D. Ridout, A realization of the Bershadsky–Polyakov algebras and their relaxed modules, Lett. Math. Phys., 111 (2021) 1–30, arXiv:2007.00396 [math.QA]] and can be interpreted as an inverse quantum hamiltonian reduction in the sense of Adamović. We use this realization to explore the representation theory of [Formula: see text] subregular W-algebras. Much of the structure encountered for [Formula: see text] and [Formula: see text] is also present for [Formula: see text]. Particularly, the simple [Formula: see text] subregular W-algebra at nondegenerate admissible levels can be realized purely in terms of the [Formula: see text] minimal model vertex algebra and [Formula: see text].