The Temperley-Lieb algebra T Ld has its origin in the study of sl2-modules: Rumer, Teller and Weyl showed (more or less) already in the 30ties that T Ld can be seen as a diagrammatic realization of the representation category of sl2-modules - providing a topological (and fun!) tool to study the latter.
In this talk I try to explain how one can proof such a realization. Our main tool is “a machine that takes dualities and produces diagrammatic categories”. In particular, we show explicitly how this “machine” works if one feeds it with q-Howe duality – which produces diagrammatic presentations of categories of sln-modules akin to the Temperley-Lieb calculus.
As an application, I give a diagrammatic version of a symmetry of HOMFLY-PT polynomials.
In principal, everything in this talk is amenable to categorification, but we have to stay in the uncategorified world for the moment.