The elliptic Hall algebra and the double Dyck path algebra

TL;DR

We establish a precise bridge between the elliptic Hall algebra $\mathcal{E}_{q,t}$ and the double Dyck path algebra $\mathbb{B}_{q,t}$ by constructing a doubled framework $\mathbb{DB}_{q,t}$ and proving that the spherical subalgebra $\mathbb{1}_0\mathbb{DB}_{q,t}\mathbb{1}_0$ is isomorphic to $\mathcal{E}_{q,t}$. The positive half $\mathcal{E}_{q,t}^{>}$ embeds as $\mathbb{1}_0\mathbb{B}_{q,t}\mathbb{1}_0$, while the full algebra is recovered via the spherical subalgebra of the doubled structure; this is realized through the Drinfeld–type presentation and detailed polynomial representations. The paper also develops faithful polynomial realizations for $\mathbb{DB}_{q,t}$ and analyzes the monodromy conditions, establishing a robust correspondence between generators $e_m,f_m,\psi^{\pm}_m$ and their ÉTA analogues, including Macdonald-operator-inspired actions. These results open avenues for lifting calibrated representations between $\mathbb{B}_{q,t}$ and $\mathcal{E}_{q,t}$ and suggest extensions to broader quantum toroidal families and integral forms, with potential applications in geometry, combinatorics, and representation theory.

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.

Theorems & Definitions (110)

• Proposition 1.1
• Theorem 1.2
• Remark 1.3
• Theorem 1.4
• Corollary 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Remark 1.9
• Corollary 1.10