The elliptic Hall algebra and the double Dyck path algebra

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.