The Elliptic Hall algebra and the deformed Khovanov Heisenberg category

Abstract

We give an explicit description of the trace, or Hochschild homology, of the quantum Heisenberg category defined by Licata and Savage. 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, 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.

49 pages, numerous figures

References in corpus (1)

• Trace as an alternative decategorification functor

Cited by in corpus (7)

• On the definition of quantum Heisenberg category
• Degenerate cyclotomic Hecke algebras and higher level Heisenberg categorification
• Affine wreath product algebras
• Quantum Frobenius Heisenberg categorification
• A Categorical Quantum Toroidal Action on Hilbert Schemes
• Trace of the twisted Heisenberg Category
• Frobenius W-algebras and traces of Frobenius Heisenberg categories