Showing papers by "David Ridout published in 2023"

Weight module classifications for Bershadsky--Polyakov algebras

Abstract: The Bershadsky--Polyakov algebras are the subregular quantum hamiltonian reductions of the affine vertex operator algebras associated with $\mathfrak{sl}_3$. 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.

A Kazhdan–Lusztig Correspondence for L-32(sl3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_{-\frac{3}{2}}(\mathfr

Abstract: The abelian and monoidal structure of the category of smooth weight modules over a non-integrable affine vertex algebra of rank greater than one is an interesting, difficult and essentially wide open problem. Even conjectures are lacking. This work details and tests such a conjecture for $$L_{-\frac{3}{2}}(\mathfrak {sl}_3)$$ via a logarithmic Kazhdan–Lusztig correspondence. We first investigate the representation theory of $$\overline{\mathcal {U}}_{\textsf{i}}^H\!(\mathfrak {sl}_3)$$ , the unrolled restricted quantum group of $$\mathfrak {sl}_3$$ at fourth root of unity. In particular, we analyse its finite-dimensional weight category, determining Loewy diagrams for all projective indecomposables and decomposing all tensor products of irreducibles. Our motivation is that this category is conjecturally braided tensor equivalent to a category of $$W^0_{\!A_2}(2)$$ -modules. Here, $$W^0_{\!A_2}(2)$$ is an orbifold of the octuplet vertex algebra $$W_{\!A_2}(2)$$ of Semikhatov, the latter being the natural $$\mathfrak {sl}_3$$ -analogue of the well known triplet algebra. Moreover, $$W^0_{\!A_2}(2)$$ is the parafermionic coset of the affine vertex algebra $$L_{-\frac{3}{2}}(\mathfrak {sl}_3)$$ . We formulate an explicit conjecture relating the representation theory of $$W^0_{\!A_2}(2)$$ and $$\overline{\mathcal {U}}_{\textsf{i}}^H\!(\mathfrak {sl}_3)$$ and work out the resulting structures of the corresponding $$L_{-\frac{3}{2}}(\mathfrak {sl}_3)$$ -modules. In particular, we obtain conjectural Loewy diagrams for the latter’s projective indecomposables and decompositions for the fusion products of its irreducibles. These products coincide with those recently computed via Verlinde’s formula. Finally, we give analogous results for $$W_{\!A_2}(2)$$ .