Classifying Relaxed Highest-Weight Modules for Admissible-Level Bershadsky–Polyakov Algebras

The Bershadsky-Polyakov algebras are the minimal quantum hamiltonian reductions of affine vertex associated to $\mathfrak{sl}_3$ and their simple quotients have a long history applications in conformal field theory string theory. Their representation theories therefore quite interesting. Here, we classify relaxed highest-weight modules, with finite-dimensional weight spaces, for all admissible but nonintegral levels, significantly generalising known classifications [arxiv:1005.0185, arxiv:1910.13781]. In particular, prove that $\mathsf{k}$ always rational category $\mathscr{O}$, whilst they admit nonsemisimple modules unless $\mathsf{k}+\frac{3}{2} \in \mathbb{Z}_{\ge0}$.