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.