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ToDo math Надо бы почитать: Langlands duality for representations of quantum groups
Authors: Edward Frenkel, David Hernandez
(Submitted on 25 Sep 2008 (v1), last revised 5 Mar 2011 (this version, v5))

Abstract: 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.

arxiv.org

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Langlands duality for finite-dimensional representations of quantum affine algebras
Authors: Edward Frenkel, David Hernandez
(Submitted on 3 Feb 2009 (v1), last revised 5 Mar 2011 (this version, v4))

Abstract: We describe a correspondence (or duality) between the q-characters of finite-dimensional representations of a quantum affine algebra and its Langlands dual in the spirit of q-alg/9708006 and 0809.4453. We prove this duality for the Kirillov-Reshetikhin modules and their irreducible tensor products. In the course of the proof we introduce and construct "interpolating (q,t)-characters" depending on two parameters which interpolate between the q-characters of a quantum affine algebra and its Langlands dual.

arxiv.org