A conditional algebraic proof of the logarithmic Kazhdan-Lusztig correspondence

The logarithmic Kazhdan-Lusztig correspondence is a conjectural equivalence between braided tensor categories of representations of small quantum groups and representations of certain vertex operator algebras. In this article we prove such an equivalence, and more general versions, using mainly algebraic arguments that characterize the representation category of the quantum group by quantities that are accessible on the vertex algebra side. Our proof is conditional on suitable analytic properties of the vertex algebra and its representation category. More precisely, we assume that it is a finite braided rigid monoidal category where the Frobenius-Perron dimensions are given by asymptotics of analytic characters.