University of Sydney Algebra Seminar

Daniel Tubbenhauer (Aarhus University)

Friday 13 February, 12:05-12:55pm, Place: 173

U_q(sl_n) Diagram Categories via q-Howe Duality

The Temperley-Lieb algebra TL_d is the mother of all diagram algebras. It was introduced by Temperley and Lieb in the 70s in their study of statistical mechanics. Also due to its simple presentation, it reappears nowadays in low dimensional topology, operator theory, algebraic combinatorics, and Lie Theory (to name a few). It has its origins in the study of sl_2-modules: Rumer, Teller and Weyl showed (more of less) already in the 30s that TL_d can be seen as a diagrammatic realisation of the representation category of sl_2-modules - providing a topological (and fun!) tool to study the latter. The ``real-life goal'' of this talk is to explain how one can prove such a realisation and to discuss some ``fancy cousins'' of TL_d, e.g. representation categories of sl_n-modules consisting of alternating tensors and representation categories consisting of symmetric tensors. Our main tool for all of these is q-Howe duality - either skew or symmetric. In principal, everything in this talk is amenable to categorification, but we stay in this uncategorified world since life is short.