Abstract
Given a strong 2-representation of a Kac–Moody Lie algebra (in the sense of Rouquier), we show how to extend it to a 2-representation of categorified quantum groups (in the sense of Khovanov–Lauda). This involves checking certain extra 2-relations, which are explicit in the definition by Khovanov–Lauda and, as it turns out, implicit in Rouquier’s definition. Some applications are also discussed.
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Notes
The KLR algebra is also called the quiver Hecke algebra in the literature.
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Acknowledgments
The authors would like to thank Mikhail Khovanov, Anthony Licata, Joshua Sussan and Ben Webster for helpful discussions and Masaki Kashiwara for helpful correspondences. S.C. was supported by NSF grants DMS-0964439, DMS-1101439, and the Alfred P. Sloan foundation. A.L. was supported by the NSF grants DMS-0739392, DMS-0855713, and the Alfred P. Sloan foundation.
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Cautis, S., Lauda, A.D. Implicit structure in 2-representations of quantum groups. Sel. Math. New Ser. 21, 201–244 (2015). https://doi.org/10.1007/s00029-014-0162-x
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DOI: https://doi.org/10.1007/s00029-014-0162-x