Categorification of the elliptic Hall algebra

TL;DR

This paper establishes a deep algebraic connection between the elliptic Hall algebra and the quantum Heisenberg category, providing new insights and representations for the algebra's universal extension.

Contribution

It introduces a categorification of the elliptic Hall algebra via Hochschild homology of the quantum Heisenberg category, revealing new structural insights.

Findings

Isomorphism between the central charge reduction of the elliptic Hall algebra and Hochschild homology of the quantum Heisenberg category.

Construction of large families of representations for the universal extension of the elliptic Hall algebra.

New algebraic tools linking categorification and elliptic Hall algebra structures.

Abstract

We show that the central charge $k$ reduction of the universal central extension of the elliptic Hall algebra is isomorphic to the trace, or zeroth Hochschild homology, of the quantum Heisenberg category of central charge $k$. As an application, we construct large families of representations of the universal extension of the elliptic Hall algebra.