Representation Theory of Categorified Quantum sl2

Abstract

Quantum sl2 gives rise to the Jones polynomial knot invariant. One of the insights of categorification is that this 3-dimensional picture is a shadow, the decategorification, of a 4-dimensional picture. Thus, the categorification of quantum sl2 gives rise through its representation theory to Khovanov homology, the categorification of the Jones polynomial. In the 3-dimensional picture, the algebra of Temperley-Lieb diagrams, used in the construction of the Jones polynomial, gives a graphical calculus for intertwiners of the representations of quantum sl2. We show that the algebra of Bar-Natan's dotted cobordisms, used in the construction of Khovanov homology, gives a graphical calculus for intertwiners of representations of categorified quantum sl2.