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Coloured Kac-Moody algebras

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I will present a new approach to deformations of a Kac-Moody algebra g (of it universal enveloping algebra U(g) more precisely) and of its modules. The construction is meant to be both elementary and systematic. We will start with the easiest case, that is to say with the Lie algebra 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 deformed Verma modules of sl2 can generate a deformation of the rigid monoidal category mod(g) of g-modules. Colourings provide in fact a classification of the deformations of the category mod(g). By Tannaka duality, we will then 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. Coloured Kac-Moody algebras were originally devised by the speaker to solve conjectures of Frenkel-Hernandez, related to the Langlands duality for quantum groups. We will see that two isogenic coloured Kac-Moody algebras can be interpolated by a third coloured Kac-Moody algebra, implying in particular a solution to the conjectures. We will also discuss the existence of a potential 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 evoke other possible applications of coloured Kac-Moody algebras

This talk is part of the Algebra and Representation Theory Seminar series.

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