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Categorical Relations Between Langlands Dual Quantum Affine Algebras: Exceptional Cases

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A Correction to this article was published on 05 October 2019

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Abstract

We first compute the denominator formulas for quantum affine algebras of all exceptional types. Then we prove the isomorphisms among Grothendieck rings of categories \({C_Q^{(t)} (t=1,2,3), \mathscr{C}_{\mathscr{Q}}^{(1)}}\) and \({\mathscr{C}_{\mathfrak{Q}}^{(1)}}\). These results give Dorey’s rule for all exceptional affine types, prove the conjectures of Kashiwara–Kang–Kim and Kashiwara–Oh, and provides the partial answers of Frenkel–Hernandez on Langlands duality for finite dimensional representations of quantum affine algebras of exceptional types.

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Change history

  • 05 October 2019

    In this addendum, we remove the ambiguity for roots of higher order of denominator formulas in our paper. These refinements state that there are roots of order 4, 5, 6, which is the first such observation of a root of order strictly larger than 3 to the best knowledge of the authors.

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Acknowledgement

The authors would like to thank Masaki Kashiwara for useful discussions. The authors would like to thank the anonymous referee for useful comments. T.S. would like to thank Ewha Womans University for its hospitality during his stay in June, 2017, where this work began. S.O. would like to thank The University of Queensland for its hospitality during his stay in February, 2018. This work benefited from computations using SageMath [65,66].

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  1. Ewha Womans University, 52 Ewhayeodae-gil, Daehyeon-dong, Seodaemun-gu, Seoul, South Korea

    Se-jin Oh

  2. School of Mathematics and Physics, The University of Queensland, St. Lucia, QLD, 4072, Australia

    Travis Scrimshaw

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  1. Se-jin Oh
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Correspondence to Travis Scrimshaw.

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Communicated by P. Zinn-Justin

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Se-jin Oh was partially supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea Government (MSIP) (NRF-2016R1C1B2013135). Travis Scrimshaw was partially supported by the Australian Research Council DP170102648.

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Oh, Sj., Scrimshaw, T. Categorical Relations Between Langlands Dual Quantum Affine Algebras: Exceptional Cases. Commun. Math. Phys. 368, 295–367 (2019). https://doi.org/10.1007/s00220-019-03287-w

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