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SUMMARY:Ben Elias (University of Oregon)
DTSTART:20200626T180000Z
DTEND:20200626T190000Z
DTSTAMP:20260424T133421Z
UID:GRT-2020/26
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/GRT-2020/26/
 ">Categorification of the Hecke algebra at roots of unity.</a>\nby Ben Eli
 as (University of Oregon) as part of Geometric Representation Theory confe
 rence\n\n\nAbstract\nCategorical representation theory is filled with grad
 ed additive categories (defined by generators and relations) whose Grothen
 dieck groups are algebras over $\\mathbb{Z}[q\,q^{-1}]$. For example\, Kho
 vanov-Lauda-Rouquier (KLR) algebras categorify the quantum group\, and the
  diagrammatic Hecke categories categorify Hecke algebras. Khovanov introdu
 ced 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 specializatio
 n into a $p$-dg algebra.  The $p$-dg Grothendieck group of a $p$-dg algebr
 a is automatically a module over $\\mathbb{Z}[\\zeta_{2p}]$... but it is N
 OT automatically the specialization of the ordinary Grothendieck group at 
 a root of unity!\n\nUpgrading the categorification to a $p$-dg algebra was
  done for quantum groups by Qi-Khovanov and Qi-Elias. Recently\, Qi-Elias 
 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 $p$-dg Grothendieck group? Do you get the quantum group/hecke alge
 bra at a root of unity\, or not?\nThis is a really hard question\, and cur
 rently the only techniques for establishing such a result involve explicit
  knowledge of all the important idempotents in the category. These techniq
 ues sufficed for quantum $\\mathfrak{sl}_n$ with $n \\le 3$\, but new tech
 niques are required to make further progress.\n\nAfter reviewing the theor
 y of $p$-dg algebras and their Grothendieck groups\, we will present some 
 new techniques and conjectures\, which we hope will blow your mind.\nEvery
 thing is joint with You Qi.\n
LOCATION:https://researchseminars.org/talk/GRT-2020/26/
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