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The elliptic Hall algebra and the deformed Khovanov Heisenberg category

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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.

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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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Authors and Affiliations

  1. Department of Mathematics, University of British Columbia, Vancouver, Canada

    Sabin Cautis

  2. Department of Mathematics, University of Southern California, Los Angeles, CA, USA

    Aaron D. Lauda

  3. Mathematical Sciences Institute, Australian National University, Canberra, Australia

    Anthony M. Licata

  4. Department of Mathematics, University of California, Riverside, Riverside, CA, USA

    Peter Samuelson

  5. Department of Mathematics, CUNY Medgar Evers, Brooklyn, NY, USA

    Joshua Sussan

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  1. Sabin Cautis
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  2. Aaron D. Lauda
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  3. Anthony M. Licata
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  4. Peter Samuelson
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  5. Joshua Sussan
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Correspondence to Aaron D. Lauda.

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Cautis, S., Lauda, A.D., Licata, A.M. et al. The elliptic Hall algebra and the deformed Khovanov Heisenberg category. Sel. Math. New Ser. 24, 4041–4103 (2018). https://doi.org/10.1007/s00029-018-0429-8

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