Invariants of 3-dimensional Kirby calculus and representation theory

Just as the Jones polynomial can be constructed from the Kauffman bracket, the coloured Jones polynomia l can be obtained from a coloured version of the Kauffman bracket, known as `Temperley-Lieb recoupling theory'. If the bracket is eval uated at a suitable root of unity, it not only yields an invariant of framed tangles, but also an invariant of 3-dimensional Kirby cal culus and thereby an invariant of (smooth) compact oriented 3-manifolds. Algebraically speaking, the root-of-unity evaluated bracket d efines a modular tensor category. Ever since Reshetikhin and Turaev invented this notion, there has been the question of whether these categories are representation categories of suitable algberas with extra structure. In this talk, I present the answer to this questi on and describe the algebras and their additional structure in terms of ribbon diagrams.