Logarithmic Kazhdan–Lusztig conjecture for screening subalgebras and Nichols algebras

Let $\alpha_1,\ldots,\alpha_n\in\mathbb{R}^n$ satisfy the convergence condition in the preceding theorem. Let $\mathcal{H}$ be the ambient algebra and let $\mathcal{W}\subset\mathcal{H}^{\otimes n}$ be the subalgebra defined by

$$\mathcal{W}=\bigcap_{i=1}^n\ker(\mathfrak{Z}_{\alpha_i}),$$

where $\mathfrak{Z}_{\alpha_i}$ are the screening operators. Set

$$\mathcal{C}={\rm Vect}_{\mathbb{R}^n}^{1,\sigma},\qquad X=\mathbb{C}_{\alpha_1}\oplus\cdots\oplus\mathbb{C}_{\alpha_n},\qquad \sigma(\alpha_i,\alpha_j)=e^{\pi\mathrm{i}(\alpha_i,\alpha_j)}.$$

Logarithmic Kazhdan–Lusztig conjecture. In good cases, there is an equivalence of braided tensor categories

$${\rm Rep}(\mathcal{W})\cong \mathcal{Z}_{\mathcal{C}}\bigl({\rm Rep}(\mathfrak{B}(X))(\mathcal{C})\bigr).$$

The right-hand side is the generalized quantum group associated to the Nichols algebra of screenings. The conjecture proposes a categorical correspondence between screening-defined vertex-algebraic objects and Nichols-algebraic quantum groups; the source notes that suitable finiteness conditions are required and that non-rigid examples can obstruct faithfulness. In the example involving the Feigin–Tipunin algebra, it specializes to an equivalence with representations of the quasi-quantum group $\tilde{u}_q(\mathfrak{g})$.