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Mathematische Zeitschrift

, Volume 286, Issue 1–2, pp 603–655 | Cite as

A general approach to Heisenberg categorification via wreath product algebras

  • Daniele Rosso
  • Alistair Savage
Article

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.

Keywords

Categorification Lattice Heisenberg algebra Tower of algebras Graded Frobenius superalgebra Fock space 

Mathematics Subject Classification

Primary 18D10 Secondary 17B10 17B65 19A22 

Notes

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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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California RiversideRiversideUSA
  2. 2.Department of Mathematics and StatisticsUniversity of OttawaOttawaCanada

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