Categorified U_q(sl₂)-theory using Bar-Natan’s approach

Abstract

For V the U_q(sl₂)-module which is the quantum version of the natural representation, we categorify the n-fold tensor product of V as a module over the Temperley-Lieb algebra. To do this, we use Bar-Natan's approach of categorifying the Temperley-Lieb algebra to categorify the weight spaces of the n-fold tensor product of V with the canonical basis. The standard basis and the dual standard basis is then categorified by constructing chain complexes in the homotopy category. After that, we consider chain maps between the categorified dual standard basis elements to construct a special chain complex containing all of them, on which the categorified Temperley-Lieb algebra acts trivially. This special complex is then used to construct functors that satisfy the conditions of the projection and inclusion, which compose to the Jones-Wenzl projector, on a higher level.