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Heisenberg and Kac–Moody categorification

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Abstract

We show that any Abelian module category over the (degenerate or quantum) Heisenberg category satisfying suitable finiteness conditions may be viewed as a 2-representation over a corresponding Kac–Moody 2-category (and vice versa). This gives a way to construct Kac–Moody actions in many representation-theoretic examples which is independent of Rouquier’s original approach via “control by \(K_0\).” As an application, we prove an isomorphism theorem for generalized cyclotomic quotients of these categories, extending the known isomorphism between cyclotomic quotients of type A affine Hecke algebras and quiver Hecke algebras.

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Notes

  1. In the quantum case there is one additional relation recorded just after (3.34).

  2. In [12, \(\S \)2.1], essentially finite Abelian categories were called “Schurian categories” but we will use the latter terminology for a slightly different notion.

  3. In [8] the terminology “locally Schurian” was used instead of “Schurian.”

  4. In [13], one also finds \((-)\)-bubbles which will not be needed here.

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Authors and Affiliations

  1. Department of Mathematics, University of Oregon, Eugene, OR, 97403, USA

    Jonathan Brundan

  2. Department of Mathematics and Statistics, University of Ottawa, Ottawa, ON, Canada

    Alistair Savage

  3. Department of Pure Mathematics, Perimeter Institute for Theoretical Physics, University of Waterloo, Waterloo, ON, Canada

    Ben Webster

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  1. Jonathan Brundan
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  2. Alistair Savage
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  3. Ben Webster
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Correspondence to Alistair Savage.

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J.B. supported in part by NSF Grant DMS-1700905. A.S. supported by Discovery Grant RGPIN-2017-03854 from the Natural Sciences and Engineering Research Council of Canada. B.W. supported by Discovery Grant RGPIN-2018-03974 from the Natural Sciences and Engineering Research Council of Canada. This research was also supported by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development and by the Province of Ontario through the Ministry of Research and Innovation.

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Brundan, J., Savage, A. & Webster, B. Heisenberg and Kac–Moody categorification. Sel. Math. New Ser. 26, 74 (2020). https://doi.org/10.1007/s00029-020-00602-5

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