The elliptic Hall algebra and the double Dyck path algebra

TL;DR

This paper embeds the positive part of the elliptic Hall algebra into the double Dyck path algebra, revealing structural connections and defining a double algebra to encompass the entire elliptic Hall algebra.

Contribution

It identifies the positive half of the elliptic Hall algebra as a spherical subalgebra within the double Dyck path algebra and introduces a double algebra to realize the full elliptic Hall algebra.

Findings

Positive half of elliptic Hall algebra embedded in double Dyck path algebra

Defined a double algebra to capture the entire elliptic Hall algebra

Identified generators using Ding-Iohara-Miki presentation

Abstract

We show that the positive half $\mathcal{E}_{q,t}^{>}$ of the elliptic Hall algebra is embedded as a natural spherical subalgebra inside the double Dyck path algebra $\mathbb{B}_{q,t}$ introduced by Carlsson, Mellit and the second author. For this, we use the Ding-Iohara-Miki presentation of the elliptic Hall algebra and identify the generators inside $\mathbb{B}_{q,t}$. In order to obtain the entire elliptic Hall algebra $\mathcal{E}_{q,t}$, we define a ``double'' $\mathbb{DB}_{q,t}$ of the double Dyck path algebra, together with its positive and negative subalgebras and an involution that exchanges them.