The Elliptic Hall algebra and the deformed Khovanov Heisenberg category

TL;DR

This paper explicitly describes the Hochschild homology of the quantum Heisenberg category, revealing its isomorphism to a part of the elliptic Hall algebra and connecting it to actions on symmetric functions.

Contribution

It provides an explicit description of the trace of the quantum Heisenberg category and establishes its isomorphism with a central extension of the elliptic Hall algebra.

Findings

Hochschild homology of the quantum Heisenberg category is explicitly described.

The trace algebra acts on symmetric functions consistent with previous constructions.

The sum of Hochschild homologies of positive affine Hecke algebras forms an algebra injecting into the elliptic Hall algebra.

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.