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2-Verma modules and the Khovanov–Rozansky link homologies

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Abstract

We explain how Queffelec–Sartori’s construction of the HOMFLY-PT link polynomial can be interpreted in terms of parabolic Verma modules for \({\mathfrak {gl}_{2n}}\). Lifting the construction to the world of categorification, we use parabolic 2-Verma modules to give a higher representation theory construction of Khovanov–Rozansky’s triply graded link homology.

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Notes

  1. We allow ourselves to harmlessly abuse notation here, which will payoff further ahead.

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Acknowledgements

The authors would like to thank Paul Wedrich for helpful discussions and for valuable comments on a preliminary version of this paper. We also thank Jonathan Grant for helpful discussions and comments on a preliminary version of this paper.

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Authors and Affiliations

  1. Max-Planck Institute for Mathematics, Bonn, Germany

    Grégoire Naisse

  2. Université Catholique de Louvain, Ottignies-Louvain-la-Neuve, Belgium

    Pedro Vaz

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  1. Grégoire Naisse
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  2. Pedro Vaz
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Correspondence to Pedro Vaz.

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G. Naisse was a Research Fellow of the Fonds de la Recherche Scientifique—FNRS, under Grant no. 1.A310.16 when starting working on this project. G. Naisse is grateful to the Max Planck Institute for Mathematics in Bonn for its hospitality and financial support. P. Vaz was supported by the Fonds de la Recherche Scientifique—FNRS under Grant no. J.0135.16.

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Naisse, G., Vaz, P. 2-Verma modules and the Khovanov–Rozansky link homologies. Math. Z. 299, 139–162 (2021). https://doi.org/10.1007/s00209-020-02658-7

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