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Friday, May 9, 2014

**Abstract:** Khovanov-Lauda-Rouquier (KLR) algebras arose out of attempts to categorify quantum groups; these algebras are certain Hecke algebras associated to symmetrizable Cartan matrices. Khovanov and Lauda came upon these algebras from an investigation of endomorphisms of Soergel bimodules (and related bimodules arising from cohomology of partial flag varieties), while Rouquier came across these algebras while analyzing Lusztig's construction of canonical bases in terms of perverse sheaves on certain quiver varieties. These algebras are closely related to filtered quiver varieties and quiver flag varieties in the sense that nil affine Hecke algebras are isomorphic to equivariant cohomology of $\widetilde{Rep(Q,\beta)}\times_{Rep(Q,\beta)} \widetilde{Rep(Q,\beta)}$ as algebras, where $\widetilde{Rep(Q,\beta)}\rightarrow Rep(Q,\beta)$ is the generalized Grothendieck-Springer resolution of the quiver variety $Rep(Q,\beta)$.

As categorification often leads to higher structure that are not seen at the level of the underlying object, the aim of this talk is to give an introduction to constructing categorifications of representations of Hecke algebras and quantum groups. Categorification techniques utilize some geometry, while at the same time, they are completely algebraic and computable. No background will be assumed and examples will be given.