TY - JOUR
T1 - The elliptic Hall algebra and the deformed Khovanov Heisenberg category
AU - Cautis, Sabin
AU - Lauda, Aaron D.
AU - Licata, Anthony M.
AU - Samuelson, Peter
AU - Sussan, Joshua
N1 - Publisher Copyright:
© 2018, Springer Nature Switzerland AG.
PY - 2018/11/1
Y1 - 2018/11/1
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=85052542868&partnerID=8YFLogxK
U2 - 10.1007/s00029-018-0429-8
DO - 10.1007/s00029-018-0429-8
M3 - Article
SN - 1022-1824
VL - 24
SP - 4041
EP - 4103
JO - Selecta Mathematica, New Series
JF - Selecta Mathematica, New Series
IS - 5
ER -