Hopf algebras, quantum groups and monoidal categories (UCL-ULB-VUB seminar)

by Daniel Bermudez (Universität Bonn), Victor Zhang (University of New South Wales)

Tuesday 8 Sept 2026, 14:00 → 16:15 Europe/Brussels

B/2nd floor-B.203 - Seminar room (Marc de Hemptinne (chemin du Cyclotron, 2, Louvain-la-Neuve))

14:00 - 14:50: Victor Zhang (University of New South Wales):  Diagrammatic Lusztig--Vogan Categories

 

Abstract:  The Kazhdan--Lusztig conjecture (1979) relates the decomposition numbers of complex semisimple Lie algebra representations in terms of a special change of basis in the Hecke algebra. In a subsequent proof of the conjecture, a complete diagrammatic language was developed for the categorification of the Hecke algebra, the category of Soergel bimodules, following the effort of various people throughout the 2000s and 2010s. The theory of real reductive Lie groups carries a result analogous to the KL-conjecture by Lusztig and Vogan (1983), that the Lusztig--Vogan modules over the Hecke algebra capture composition multiplicities of $(\mathfrak{g},K)$-modules. We will motivate diagrammatic tools in mathematics and present the beginnings of a diagrammatic theory for a categorification of these Lusztig--Vogan modules.

 

15:10 - 16:00: Daniel Bermudez (Universität Bonn):  The A_infty Drinfeld centralizer as a homotopy limit

 

Abstract: By a theorem by Street, the Drinfeld center can be described as a bilimit of a (truncated) cosimplicial diagram. I will explain how to generalize this picture to dg-categories. Elias and Hogancamp recently introduced an A_infty centralizer of a monoidal dg-category, defined through explicit data. I will show that this centralizer arises as a homotopy limit of a cosimplicial diagram built from A_infinity functor categories.  As a consequence, we conclude that the A_infty centralizer is a concrete dg-model for Lurie's infty-categorical centralizer on the infty-category of dg-categories, and thus can be inherited with an E2-algebra structure, i.e. a braiding.