ALGB Seminar: Loïc Poulain d’Andecy (Université de Reims)

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

Abstract: (joint work with Nicolas Crampé)
The symmetric group (and the Hecke algebra) appear in the classical Schur--Weyl duality, which explains how to understand the centraliser of tensor products of the vector representation of U(sl_N) (and of U_q(sl_N)). 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(sl_N) and U_q(sl_N) (the "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 representation theory, some motivations come from mathematical physics (for the Yang--Baxter equation) and from low-dimensional topology (quantum invariants of knots and links) and I will try to indicate them along the way.