Abstract
We associate a monoidal category \({\mathcal {H}}_B\), defined in terms of planar diagrams, to any graded Frobenius superalgebra B. This category acts naturally on modules over the wreath product algebras associated to B. To B we also associate a (quantum) lattice Heisenberg algebra \({\mathfrak {h}}_B\). We show that, provided B is not concentrated in degree zero, the Grothendieck group of \({\mathcal {H}}_B\) is isomorphic, as an algebra, to \({\mathfrak {h}}_B\). For specific choices of Frobenius algebra B, we recover existing results, including those of Khovanov and Cautis–Licata. We also prove that certain morphism spaces in the category \({\mathcal {H}}_B\) contain generalizations of the degenerate affine Hecke algebra. Specializing B, this proves an open conjecture of Cautis–Licata.
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Acknowledgements
The authors would like to thank A. Licata and S. Cautis for sharing their preliminary notes, based on conversations with M. Khovanov, regarding Heisenberg categorification depending on symmetric Frobenius algebras. They would also like to thank J. Brundan, A. Licata, A. Ram, J. Sussan, and O. Yacobi for useful conversations.
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The second author was supported by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada.
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Rosso, D., Savage, A. A general approach to Heisenberg categorification via wreath product algebras. Math. Z. 286, 603–655 (2017). https://doi.org/10.1007/s00209-016-1776-9
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DOI: https://doi.org/10.1007/s00209-016-1776-9