Categorification of the Hecke algebra at roots of unity. Speaker(s): Ben Elias
Abstract: Categorical representation theory is filled with graded additive categories (defined by generators and relations) whose Grothendieck groups are algebras over mathbb{Z}[q,q^{1}]. For example, KhovanovLaudaRouquier (KLR) algebras categorify the quantum group, and the diagrammatic Hecke categories categorify Hecke algebras. Khovanov introduced Hopfological algebra in 2006 as a method to potentially categorify the specialization of these mathbb{Z}[q,q^{1}]algebras at q = zeta_n a root of unity. The schtick is this: one equips the category (e.g. the KLR algebra) with a derivation d of degree 2, which satisfies d^p = 0 after specialization to characteristic p, making this specialization into a pdg algebra. The pdg Grothendieck group of a pdg algebra is automatically a module over mathbb{Z}[zeta_{2p}]... but it is NOT automatically the specialization of the ordinary Grothendieck group at a root of unity!
Upgrading the categorification to a pdg algebra was done for quantum groups by QiKhovanov and QiElias. Recently, QiElias accomplished the task for the diagrammatic Hecke algebra in type A, and ruled out the possibility for most other types. Now the question is: what IS the pdg Grothendieck group? Do you get the quantum group/hecke algebra at a root of unity, or not?
This is a really hard question, and currently the only techniques for establishing such a result involve explicit knowledge of all the important idempotents in the category. These techniques sufficed for quantum mathfrak{sl}_n with n le 3, but new techniques are required to make further progress.
After reviewing the theory of pdg algebras and their Grothendieck groups, we will present some new techniques and conjectures, which we hope will blow your mind.
Everything is joint with You Qi.
Date: 26/06/2020  2:00 pm
Collection: Geometric Representation Theory
