A categorification of quantum sl(2)

sl3-Foams and the Khovanov-Lauda categorification of quantum slk.

sl 3-Foams and the Khovanov-Lauda categorification of quantum sl k. Abstract In this paper I define certain interesting 2-functors from the Khovanov-Lauda 2-category which categorifies quantum sl k , for any k > 1, to a 2-category of universal sl 3 foams with corners. For want of a better name I use the term foamation to indicate those 2-functors. I conjecture the existence of similar 2-functor...

Categorification of the Jones-wenzl Projectors

The Jones-Wenzl projectors pn play a central role in quantum topology, underlying the construction of SU(2) topological quantum field theories and quantum spin networks. We construct chain complexes Pn , whose graded Euler characteristic is the “classical” projector pn in the Temperley-Lieb algebra. We show that the Pn are idempotents and uniquely defined up to homotopy. Our results fit within ...

N ov 2 00 6 Categorification of level two representations of quantum sl n via generalized arc rings

POSITIVITY AND THE CANONICAL BASIS OF TENSOR PRODUCTS OF FINITE-DIMENSIONAL IRREDUCIBLE REPRESENTATIONS OF QUANTUM sl(k)

In a categorification of tensor products of fundamental representations of quantum sl(k) via highest weight categories, the indecomposable tilting modules descend to the canonical basis. Projective functors map tilting modules to tilting modules implying the coefficients of the canonical basis of tensor products of finite dimensional, irreducible representations under the action of the Chevalle...

Knot Homology via Derived Categories of Coherent Sheaves I, Sl(2) Case

Using derived categories of equivariant coherent sheaves, we construct a categorification of the tangle calculus associated to sl(2) and its standard representation. Our construction is related to that of Seidel-Smith by homological mirror symmetry. We show that the resulting doubly graded knot homology agrees with Khovanov homology.

Matrix Factorizations and Colored Moy Graphs

We assign matrix factorizations to colored MOY graphs, which gives a natural generalization of the graph homology defined by Khovanov and Rozansky in [8]. We also discuss some basic properties of this generalization. This work is a first step toward categorification of the colored MOY invariant, which is equivalent to quantum sl(N)-invariant for links whose components are colored by exterior pr...