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SUMMARY:Coloured Kac-Moody algebras - Alexandre Bouayad (University of Cam
 bridge)
DTSTART:20141105T163000Z
DTEND:20141105T173000Z
UID:TALK55507@talks.cam.ac.uk
CONTACT:David Stewart
DESCRIPTION:I will present a new approach to deformations of a Kac-Moody a
 lgebra g (of it universal enveloping algebra U(g) more precisely) and of i
 ts modules. The construction is meant to be both elementary and systematic
 .\nWe will start with the easiest case\, that is to say with the Lie algeb
 ra sl2\, by colouring the crystal graphs of the Verma modules of sl2 with 
 deformations of the natural numbers. We will explain how a given colouring
  defines a deformation of the Verma modules of sl2. We will see how deform
 ed Verma modules of sl2 can generate a deformation of the rigid monoidal c
 ategory mod(g) of g-modules. Colourings provide in fact a classification o
 f the deformations of the category mod(g). By Tannaka duality\, we will th
 en obtain that colourings form a groupoid\, isomorphic to the groupoid of 
 deformations of the Hopf algebra U(g). We will retrieve in particular the 
 Drinfeld-Jimbo quantum algebra from a colouring by q-numbers.\nColoured Ka
 c-Moody algebras were originally devised by the speaker to solve conjectur
 es of Frenkel-Hernandez\, related to the Langlands duality for quantum gro
 ups. We will see that two isogenic coloured Kac-Moody algebras can be inte
 rpolated by a third coloured Kac-Moody algebra\, implying in particular a 
 solution to the conjectures. We will also discuss the existence of a poten
 tial link between coloured Kac-Moody algebras and quiver Hecke algebras of
  Khovanov-Lauda-Rouquier. This may pave the way to a categorification and 
 to a geometric realisation of coloured Kac-Moody algebras\, with eventual 
 connections to the geometric Langlands correspondence. We will lastly evok
 e other possible applications of coloured Kac-Moody algebras
LOCATION:MR12
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