Langlands duality for representations of quantum groups

Details Langlands duality for representations of quantum groups Langlands duality for representations of quantum groups

2008-09-25

arxiv.org/abs/0809.4453v5

Math. Ann. 349 (2011), no. 3, 705--746

Mathematics

Quantum Algebra

37 pages. References added. Accepted for publication in Mathematische Annalen

Scientific paper

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.

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