The elliptic Hall algebra and the deformed Khovanov Heisenberg category

Sabin Cautis, Aaron D. Lauda*, Anthony M. Licata, Peter Samuelson, Joshua Sussan

*Corresponding author for this work

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    9 Citations (Scopus)

    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 σ= σ¯ - 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 AHn+ 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.

    Original languageEnglish
    Pages (from-to)4041-4103
    Number of pages63
    JournalSelecta Mathematica, New Series
    Volume24
    Issue number5
    DOIs
    Publication statusPublished - 1 Nov 2018

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