Web calculi in representation theory

A striking question is if one can present the category Mod(g) of finite-dimensional representations of some Lie algebra g, via generators and relations – or even better: via diagrammatic generators and relations.

The question itself is very hard and only partial solutions are known: it goes back to work of Schur and Brauer that the subcategory of Mod(g) tensor generated by the vector representation can be (almost) described if g is of classical type and certain dimensions are “big enough”. Moreover, Rumer, Teller and Weyl showed (more or less) already in the 30ties that the Temperley-Lieb algebra can be seen as a diagrammatic realization of the representation category of sl2-modules tensor generated by the vector representation of sl2 – providing a topological (and fun!) tool to study the later.

In this talk I explain “howe” one can prove such a (diagrammatic) realization. Everything in this talk can be quantized (roughly: put quantum brackets everywhere) and everything is amenable to categorification, but it is summer and I will keep it as easy as possible.