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Selecta Mathematica

, Volume 24, Issue 5, pp 4041–4103 | Cite as

The elliptic Hall algebra and the deformed Khovanov Heisenberg category

  • Sabin Cautis
  • Aaron D. Lauda
  • Anthony M. Licata
  • Peter Samuelson
  • Joshua Sussan
Article
  • 17 Downloads

Abstract

We give an explicit description of the trace, or Hochschild homology, of the quantum Heisenberg category defined in  Licata and Savage (Quantum Topol 4(2):125–185, 2013. arXiv:1009.3295). We also show that as an algebra, it is isomorphic to “half” of a central extension of the elliptic Hall algebra of Burban and Schiffmann (Duke Math J 161(7):1171–1231, 2012. arXiv:math/0505148), specialized at \(\sigma = {\bar{\sigma }}^{-1} = q\). A key step in the proof may be of independent interest: we show that the sum (over n) of the Hochschild homologies of the positive affine Hecke algebras \(\mathrm{AH}_n^+\) is again an algebra, and that this algebra injects into both the elliptic Hall algebra and the trace of the q-Heisenberg category. Finally, we show that a natural action of the trace algebra on the space of symmetric functions agrees with the specialization of an action constructed by Schiffmann and Vasserot using Hilbert schemes.

Mathematics Subject Classification

81R10 20C08 17B65 18D10 

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Notes

Acknowledgements

The authors are grateful to B. Elias, F. Goodman, S. Morrison, O. Schiffmann and D. Tubbenhauer for helpful conversations. The authors are grateful to the referee for pointing out Remark 5.3. S.C. was supported by an NSERC discovery/accelerator Grant. A.D.L. was partially supported by NSF Grant DMS-1255334 and by the Simons Foundation. A.M.L. was supported by an Australian Research Council Discovery Early Career fellowship. J.S. was supported by NSF Grant DMS-1407394, PSC-CUNY Award 67144-0045, and an Alfred P. Sloan Foundation CUNY Junior Faculty Award.

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Sabin Cautis
    • 1
  • Aaron D. Lauda
    • 2
  • Anthony M. Licata
    • 3
  • Peter Samuelson
    • 4
  • Joshua Sussan
    • 5
  1. 1.Department of MathematicsUniversity of British ColumbiaVancouverCanada
  2. 2.Department of MathematicsUniversity of Southern CaliforniaLos AngelesUSA
  3. 3.Mathematical Sciences InstituteAustralian National UniversityCanberraAustralia
  4. 4.Department of MathematicsUniversity of California, RiversideRiversideUSA
  5. 5.Department of MathematicsCUNY Medgar EversBrooklynUSA

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