Templerley-Lieb algebra at roots of unity

Speaker: Jieru Zhu 
Affiliation: University of Queensland

Abstract

The well-known Temperley-Lieb (TL) algebras are defined over a field containing a quantum variable. When this variable is a root of unity, the TL algebra is no longer semisimple, and its representations are recently studied from the perspective of cellular algebras. In joint work with Benkart, Biswal, Kirkman, Nguyen, we show that the TL algebra is isomorphic to the centralizer algebra of the Drinfeld double of the Taft algebra, a quasitriangular Hopf algebra with a distinguished ribbon element. In joint work with Sutton, Tubbenhauer, Wedrich, we use TL diagrams to give a quiver presentation for the tilting category for SL_2, over a field of positive characteristic and containing a root of unity. We also derived recursive formulas and traces for the modular Jones-Wenzl idempotents. Current work involves studying modular Schur-Weyl duality as well as its representations via Young tableaux combinatorics.