Langlands duality for representations of quantum groups

Edward Frenkel , David Hernandez (1, 2)

1 LMV - Laboratoire de Mathématiques de Versailles
2 DMA - Département de Mathématiques et Applications - ENS Paris

We establish a correspondence (or duality) between the characters and the crystal bases of finite-dimensional representations of quantum groups associated to Langlands dual semi-simple Lie algebras. This duality may also be stated purely in terms of semi-simple Lie algebras. To explain this duality, we introduce an "interpolating quantum group" depending on two parameters which interpolates between a quantum group and its Langlands dual. We construct examples of its representations, depending on two parameters, which interpolate between representations of two Langlands dual quantum groups.

Domaines: Algèbres quantiques [math.QA]; Théorie des représentations [math.RT]

Edward Frenkel, David Hernandez. Langlands duality for representations of quantum groups. 2008. ⟨hal-00325608⟩