Proving the Logarithmic Kazhdan-Lusztig Correspondence

Abstract: The logarithmic Kazhdan-Lusztig correspondence by B. Feigin and others is a conjectural equivalence between braided tensor categories of representations of quantum groups and of certain vertex algebras, which are algebras with an analytic flavour that appear in quantum field theory. I will give a gentle introduction into the physics side and recall some previous result of mine that certain analytic operators called screenings fulfill the relations of an associated Nichols algebra.

In arXiv:2501.10735 I recently gave a proof of the conjectural category equivalence in quite general situations, also including Nichols algebras beyond quantum groups, under the assumption that the vertex algebra side is analytically nice enough. The proof is based on joint work with T. Creutzig and M. Rupert, in which we settled first small cases. The proof is almost completely algebraic and interesting in its own right, the essential statement is: Every braided tensor category together with a big commutative algebra A, such that the category of local A-modules is semisimple and the category of A-modules contains no additional simple modules, is equivalent to representations of a quantum group associated to a Nichols algebra, which is determined by certain Ext1-groups. In a certain sense, this is a categorical and braided version of the Andruskiewitsch-Schneider program, and prominently uses important results in this area by I. Angiono and others.

Mathematics

Audience: researchers in the topic

European Quantum Algebra Lectures (EQuAL)

Series comments: EQuAL is an online seminar series on quantum algebra and related topics such as topological and conformal field theory, operator algebra, representation theory, quantum topology, etc. sites.google.com/view/equalseminar/home