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Monoidal categorification and quantum affine algebras II

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Abstract

We introduce a new family of real simple modules over the quantum affine algebras, called the affine determinantial modules, which contains the Kirillov-Reshetikhin (KR)-modules as a special subfamily, and then prove T-systems among them which generalize the T-systems among KR-modules and unipotent quantum minors in the quantum unipotent coordinate algebras simultaneously. We develop new combinatorial tools: admissible chains of \(i\)-boxes which produce commuting families of affine determinantial modules, and box moves which describe the T-system in a combinatorial way. Using these results, we prove that various module categories over the quantum affine algebras provide monoidal categorifications of cluster algebras. As special cases, Hernandez-Leclerc categories \(\mathscr{C}_{{\mathfrak{g}}}^{0}\) and \(\mathscr{C}_{{\mathfrak{g}}}^{-}\) provide monoidal categorifications of the cluster algebras for an arbitrary quantum affine algebra.

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Acknowledgements

The second, third and fourth authors gratefully acknowledge for the hospitality of RIMS (Kyoto University) during their visit in 2020.

Funding

The research of M. Kashiwara was supported by Grant-in-Aid for Scientific Research (B) 23K20206, Japan Society for the Promotion of Science.

The research of M. Kim was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea government(MSIT) NRF-2022R1F1A1076214 and NRF-2020R1A5A1016126.

The research of S.-j. Oh was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea government(MSIT) (NRF-2022R1A2C1004045).

The research of E. Park was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea Government(MSIT)(RS-2023-00273425 and NRF-2020R1A5A1016126).

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Authors and Affiliations

  1. Kyoto University Institute for Advanced Study, Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502, Japan

    Masaki Kashiwara

  2. Korea Institute for Advanced Study, Seoul, 02455, Korea

    Masaki Kashiwara

  3. Department of Mathematics, Kyung Hee University, Seoul, 02447, Korea

    Myungho Kim

  4. Department of Mathematics, Sungkyunkwan University, Suwon, South Korea

    Se-jin Oh

  5. Department of Mathematics, University of Seoul, Seoul, 02504, Korea

    Euiyong Park

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  1. Masaki Kashiwara
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  2. Myungho Kim
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  4. Euiyong Park
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Correspondence to Se-jin Oh.

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Kashiwara, M., Kim, M., Oh, Sj. et al. Monoidal categorification and quantum affine algebras II. Invent. math. 236, 837–924 (2024). https://doi.org/10.1007/s00222-024-01249-1

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