Séminaires : Séminaire d'Algèbre

Abstract: As a part of Langlands duality, certain equations were found in two different areas of mathematics. They are known as Baxter’s TQ systems and the QQ type systems, as they trace back to Baxter’s study on integrable models in the 1970s. During the same decade, similar systems of equations were discovered in the area of ordinary differential equations (ODE) by Sibuya, Voros and others. Today, this remarkable correspondence is realized as a duality between representation theory of nontwisted quantum affine algebras (QAA) and the theory of opers for their Langlands dual Lie algebras.
We are interested in this duality when the roles of the affine Lie algebra and its dual are exchanged. When the nontwisted QAA is of type BCFG, its dual will be a twisted QAA. To exchange their roles amounts to studying representations of twisted QAAs.
In this talk, we will begin by reviewing this story. We will explain the representation theory of twisted QAAs and their Borel algebras. We will explain the expected relationship between twisted and nontwisted types, and we will establish TQ systems and QQ^{~} systems for twisted QAAs.

This talk will take place in hybrid mode at the IHP.