Loïc Poulain d'Andecy, Laboratoire de Mathématiques de Reims, Université de Reims Champagne-Ardenne

Centralisers of tensor representations of classical and quantum sl(N)

The well-known Hecke algebra appears in the Schur--Weyl duality, which explains how to understand the centraliser of tensor products of the natural representation of U_q(sl_N) (the "spin 1/2" representation if N=2). In this talk, I will start by quickly reviewing this classical statement. Then I will introduce a class of new algebras in order to consider other representations of U_q(sl_N) ("higher spin" representations if N=2). I will explain how to construct them explicitly, I will fully describe their representation theory, and I will give a diagrammatic presentation of them resulting in a complete description of the centralisers. These algebras generalise the symmetric group and more generally the Hecke algebras and the Temperley--Lieb algebras. Aside from pure representation theory, some motivations come from mathematical physics (Yang--Baxter equation) and from low-dimensional topology (quantum invariants of knots and links) and I will try to indicate them along the way. This is joint work with Nicolas Crampé.