OIST Representation Theory and Algebraic Combinatorics Unit

A Nichols algebra can be associated with an object of a braided tensor category through braid-group combinatorics or a universal characterization. Besides reconstructing quantum groups from their Cartan components, these algebras can generate new braided tensor categories with rich root-system structures. Simon Lentner connects this construction to conformal field theory, where braidings arise from monodromy of differential equations such as the Knizhnik–Zamolodchikov equation. The Kazhdan–Lusztig correspondence relates the representation theories in these settings to quantum groups. The talk describes a nonsemisimple form of this correspondence, conjectured by Feigin and collaborators and proved using Nichols algebras and categorical methods. It also presents recent work identifying the braided tensor category of affine sl(2) modules at admissible level through a quantum group of type sl(2|1).