Marco Mackaay (Universidade do Algarve)

Categorifications of quantum groups, Hecke algebras and q-Schur algebras

For any Cartan datum, Khovanov and Lauda defined a diagrammatic 2-category whose Grothendieck algebra they conjectured to be isomorphic to the associated quantum group. For A_n, they proved their conjecture. For any simple Lie algebra, Soergel defined a monoidal category of bimodules whose Grothendieck algebra he proved to be the associated Hecke algebra. Elias and Khovanov gave a diagrammatic version of Soergel's monoidal category for the A_n series.

In my talk I will sketch these results for the series A_n and show how they are related via a quotient of the Khovanov-Lauda 2-category, whose Grothendieck algebrais isomorphic to the q-Schur algebra. This is joint work with M. Stosic and P. Vaz.