The tensor category of representations of quantum sln has a graphical description in terms of webs, which represent the intertwiners. Moreover, there exists a presentation in terms of generating webs and relations. For sl2 this is old stuff (1970s) based on work by Temperley-Lieb (for sl2 the webs are particularly simple, they are composed of just arcs and circles), for sl3 this was worked out by Kuperberg (1997), but only recently Cautis-Kamnitzer-Morrison (2012) obtained a complete presentation in terms of generating webs and relations for general sln.
In Khovanov's seminal work on the categorification of the Jones polynomial, he also defined a new algebra, which he called the "arc algebra". Its definition uses arcs and cobordisms between them, modulo a set of relations. Among other things he proved that the Grothendieck group of an arc algebra is isomorphic to a certain space of invariant sl2 intertwiners and that there is a natural set of idempotents in the algebra which corresponds to the dual canonical basis under this isomorphism. The arc algebras have many other interesting properties, e.g. they are related to the geometry 2-row Springer varieties, which were worked out by a variety of people, e.g. Chen-Khovanov, Stroppel, Brundan-Stroppel, Stroppel-Webster.
In my talk I will explain how to define the sln analogue of the arc algebras, which I call the sln web algebras, and explain the analogues of the results for arc algebras which I sketched above. For sl3 this was done in joint work with Weiwei Pan and Daniel Tubbenhauer. For general sln this was done in joint work with Yasuyoshi Yonezawa.
